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Acyclic Coloring of Graphs
Submitted in partial fulfillment of
the requirements for the degree of
Master of Science (by Research)
in
Computer Science
by
Bharat Joshi
200807031
Center for Security, Theory and Algorithmic Research
International Institute of Information Technology
Hyderabad, INDIA
February, 2010
c©Copyright by Bharat Joshi, 2010
INTERNATIONAL INSTITUTE OF INFORMATION TECHNOLOGY
Hyderabad, India
CERTIFICATE
It is certified that the work contained in this thesis, titled “Acyclic Coloring of Graphs” by Bharat Joshi,
has been carried out under my supervision and is not submitted elsewhere for a degree.
Date Advisor: Dr. Kothapalli
To Mum, Dad and Vicky
Acknowledgements
I am grateful to my advisor Dr. Kishore Kothapalli for introducing me to research and believing that
I could do good research. I also appreciate his patience for bearing me through all the missed meetings
and untimely reports.
I would also like to thank all the faculty that I worked with and many students including my class-
mates, seniors and juniors from whom I have learnt immensely.
I would specially like to thank my friends Ankur Handa, Sandeep Saini, Sandeep Kumar, Gaurav
Ruhela, Maneesh Upmanyu, Pratyush Bhatt, Supreeth Achar, Sumit Maheshwari, Anshul Agarwal and
Rishabh Mukherjee for all the support, encouragement and intellectual capital they provided me with.
I also acknowledge the support of Department of Science and Technology, Government of India,
New Delhi, via project no: SR/S4/MS:406/06.
Above all I am thankful to my family for their patience and support.
Abstract
Graph coloring is a fundamental problem in Computer Science. Despite its status as a computation-
ally hard problem, it is still an active area of research. Depending on the specific area of requirement or
an application, there are several variants of coloring.
One such variant of graph coloring is Acyclic Graph Coloring. This requires that after a graph has
been colored there should not exist any cycle that runs on vertices that are colored by only two colors.
Such a cycle is called a bichromatic cycle. When a coloring does not present itself with a bichromatic
cycle, it is said to have acyclically colored the given graph.
As with the case of proper coloring of graphs, acyclic coloring can either be on vertex set or the edge
set of a graph. When such a coloring is performed on the edges of a graph, a bichromatic cycle is said
to be one that runs on the edges that are colored by only two colors. In absence of such a bichromatic
cycle, an edge coloring is said to be an acyclic edge coloring of that graph.
The smallest number of colors required to acyclically vertex color a graph is called its acyclic chro-
matic number and is denoted by a(G). Similarly, the smallest number of colors required to acyclically
edge colors a graph is called its acyclic edge chromatic number and is denoted by a′(G).
There are several applications of acyclic coloring of a graph which include Hessian computation,
applications in coding theory as well as other theoretical problems such as Minimim Feedback Vertex
Set. In addition to this, acyclic chromatic numbers are closely related to several other types to colorings
of a graph each of which are applicable in several other fields of interest. A notable example among
these is the star chromatic number of a graph and star arboricity. Star graphs are used to model star
network topologies that have applications in computer networks and distributed computing.
The problems of finding the acyclic chromatic number or acyclic edge chromatic number of a graph
are known be NP-hard. Also, there is a very little knowledge about such a characteristic of a graph even
vi
vii
for the most simple classes of graphs like complete graphs.
This thesis is dedicated to explore some common classes of graphs for which the acyclic chromatic
number or the acyclic edge chromatic number can be found out. We also try to strengthen our results by
providing contructive bounds and efficient linear time algorithms for the graphs that we study. In this
respect, we have presented appreciable results for acyclic edge chromatic number of complete graphs
when the number of vertices can be represented as a certain product of a prime and numbers one less
than a prime.
Another class of graphs that we have considered are the grid-like graphs. The name implies that
these graphs are structurally similar to a standard grid graph. More specifically, these graphs can be
represented as a product of certain simple class of graphs such as a path or a cycle. In this case, we have
successfully acyclically vertex colored some specific cases of these graphs.
Lastly, we mention here that through out this work, we have provided elaborate intuition on how
we arrived at our coloring schemes and how such schemes are helpful in gaining an understanding
of the structure of the graphs in question. This intuition can be very helpful for gaining an overall
understanding of this work apart from the mathematical proofs. It should also be an important asset in
any possible extension of our work.
Contents
Chapter Page
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Mathematical Definition of a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Complete Graphs and Complete Bipartite Graphs . . . . . . . . . . . . . . . . . . . . 2
1.3 Proper Vertex/Edge Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Acyclic Vertex/Edge Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5.1 Applications to Other Colorings . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5.2 Automatic Differentiation and Hessian Computation . . . . . . . . . . . . . . 6
1.5.3 Minimum Feedback Vertex Cover . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5.4 Coding Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Background and Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Greedy Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 Examining the Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.3 Random Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.4 Integer Programming Formulation . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Computational Status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 One-factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2 Perfect 1-factorization Conjecture . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Probabilistic Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4.1 Lovasz Local Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.2 Alon et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Degree based Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Coloring Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1 Alon’s Scheme for Kp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Alon’s Scheme for Kp−1,p−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Complete bipartite graph K(p−1)(q−1),(p−1)(q−1) . . . . . . . . . . . . . . . . . . . . . 17
3.4 Complete graph Kp(q−1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 Complete graph Kp(q−1)(r−1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.6 Extension and Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
viii
CONTENTS ix
4 Grid like Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1 d-dimensional Partial Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Graph Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 Acyclic Edge Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4 Acyclic Vertex Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.5 Colorings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.5.1 CYCLES 2 PATHS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.5.1.1 Cm2P2,m is odd . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.5.1.2 Cm2P2,m is even . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.5.1.3 Cm2Pk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.5.1.4 Optimality of coloring of Cm2Pk . . . . . . . . . . . . . . . . . . . 38
4.5.2 CYCLES 2 CYCLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.5.2.1 Cm2Ck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2 Directions for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2.1 Edge Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2.2 Vertex Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Related Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
List of Figures
Figure Page
1.1 A complete graph on 7 vertices, K7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 A bipartite graph with independent sets of size 5 and 4 . . . . . . . . . . . . . . . . . 3
1.3 A complete bipartite graph with independent sets of size 3 and 2, K3,2 . . . . . . . . . 3
3.1 A complete bipartite graph K8,8, viewed as K(3−1)(5−1),(3−1)(5−1) . . . . . . . . . . . 17
3.2 A complete graph on 10 vertices, K10, viewed as K5(3−1) . . . . . . . . . . . . . . . . 22
3.3 A complete graph on 5 multi-nodes of Kpq for some primes p, q greater than 2 . . . . . 27
4.1 A Cm2P2, where m is odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 A Cm2P2, where m is even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3 A Cm2P3 colored with 6 colors, where m is odd . . . . . . . . . . . . . . . . . . . . 37
4.4 A Cm2P3 colored with 4 colors, where m is odd . . . . . . . . . . . . . . . . . . . . 37
4.5 All possible distinct color combinations for C42P2 using 3 colors . . . . . . . . . . . 39
4.6 A Cm2P3 colored with 7 colors, where m is odd . . . . . . . . . . . . . . . . . . . . 41
x
Chapter 1
Introduction
A graph is a mathematical entity which represents relationships between certain objects of interest.
For instance, a graph could represent a set of people as its vertex set in which every edge connects two
persons if they know each other. Using this relationship graph, we can find out certain properties about
this set of people, like which set of persons forms a well connected group or how many persons in this
set do not know a certain group of people etc. Either vertices or edges of a graph can be colored to
show various properties of that graph. Graph coloring has been a long studied problem, and has many
applications resulting in several variants of it.
1.1 Mathematical Definition of a Graph
Before moving further, let us first formally define what a graph is.
A graph G is denoted by G = (V,E) where V is the set of objects of concern, called vertices, and, E
is the set of relationships between vertices in the set V , called edges. An edge is generally represented
in the form (u, v), which means that it connects the vertices u and v, where u, v ∈ V . An edge is said
to be undirected if the relationsip represented by it is symmetric like the relationship graph described
above or directed if the relationship is not symmetric, like family relations, father-son etc.
The number of vertices connected to a given vertex is called its degree. The degree of a graph is the
maximum over the degrees of its vertices, and is denoted by ∆(G), or simply ∆. An undirected graph is
called simple if it does not contain any multi-edges or self loops. Unless otherwise stated, all the graphs
in this text are considered to be simple.
1
1.2 Complete Graphs and Complete Bipartite Graphs
A complete graph is a graph in which every vertex v ∈ V is connencted to every other vertex
u ∈ V, u 6= v. In other words, a complete graph has an edge set E such that E = (u, v)|u 6=
v and u, v ∈ V .
A complete graph on n vertices has n(n − 1)/2 edges and is denoted by Kn. It is a regular graph of
degree n − 1.
Figure 1.1 A complete graph on 7 vertices, K7
A bipartite graph is a graph whose vertex set can be partitioned into two disjoint sets U and V such
that every edge is between a vertex in U to a vertex in V , i.e. U and V are independent sets. Equivalen-
tely, a graph is bipartite if and only if it does not contain an odd cycle. An independent set is a set of
vertices in which neither of the vertices is connected to any other vertex in the set.
A complete bipartite graph is a bipartite graph G = (U ∪ V,E) where every vertex in the set U has
an edge connecting it to every vertex in the set V , i.e. E = (u, v)|u ∈ U and v ∈ V . A complete
bipartite graph with independent sets of size m and n, is denoted by Km,n.
1.3 Proper Vertex/Edge Coloring
The vertices or edges of a graph can be assigned colors subject to certain constraints. Such a color-
ing generally represents some property exhibited by that graph. For example, the edges of relationship
graph from the introductory section can be colored with red color if persons connencted by that edge
2
Figure 1.2 A bipartite graph with independent sets of size 5 and 4
Figure 1.3 A complete bipartite graph with independent sets of size 3 and 2, K3,2
are a married couple and with blue color otherwise.
In case of vertex coloring, such a coloring is said to be proper if no two adjacent vertices have been
assigned the same color. A corresponding definition for an edge coloring to be proper requires that no
two edges incident at the same vertex can be assigned the same color. Unless otherwise stated, a vertex
or edge coloring is generally assumed to be proper.
A coloring of a graph using at most k colors is called a k-coloring of that graph. The smallest number
of colors needed to color the vertices of a graph G is called its chromatic number, and, is denoted by
χ(G). A graph G is called k-chromatic if χ(G) = k.
A subset of vertices assigned the same color is called a color class. When a graph G is colored with
χ(G) colors, every color class forms an independent set. Thus, the vertex set of a k-chromatic graph
3
can be partitioned into k independent sets.
The smallest number of colors needed to color the edges of a graph G is called its edge chromatic
number, and, is denoted by χ′(G). Similar to vertex coloring, when edge colored with χ′(G) colors,
the edge set of a graph G is partitioned into k matchings. A matching is a set of edges without common
vertices.
1.4 Acyclic Vertex/Edge Coloring
A coloring of a graph is said to register a bichromatic cycle if there exists a cycle in the graph such
that all the vertices (or edges in case of edge coloring) in that cycle are colored with only two colors.
Note that, a bichromatic cycle can only be of even length, owing to properness of coloring.
A vertex or edge coloring of a graph is said to be acyclic if there does not exist any bichromatic
cycle. In other words, the subgraph induced by any two color classes is acyclic. That is, it is a disjoint
collection of trees, or a forest.
The smallest number of colors needed to acyclically color the vertices of a graph is called its acyclic
chromatic number, and, is denoted by a(G). Similarly, the acyclic edge chromatic number is denoted
by a′(G).
1.5 Applications
Acyclic coloring has many applications. Several problems get reduced into acyclic coloring. We
present some examples here.
1.5.1 Applications to Other Colorings
Acyclic coloring is related to many other colorings. Specifically, knowing the acyclic chromatic
number can help determine bounds on other colorings in terms of it. Following are some notable exam-
ples.
For a family of graphs G and an integer n ≥ 1, let λ(G,n) denote the minimum possible number of
vertices in an edge colored graph H such that each member of G that is n-colorable maps homomorphi-
4
cally into it. If Gk is the family of all graphs with acyclic chromatic number not exceeding k, then, for
every k and n, λ(Gk, n) ≤ knk−1 [AM98].
A star graph on k + 1 vertices, is a complete bipartite graph K1,k, and, is denoted by Sk. A star
Coloring of a graph G is a vertex coloring in which every path on four vertices uses at least three distinct
colors. Equivalently, the induced subgraphs formed by the vertices of any two color classes have con-
nected components that are star graphs. It is shown that the star chromatic number, χS(G) of a graph
with acyclic chromatic number k is not greater than k2k−1 [Gru73].
An oriented graph is a simple directed graph such that for every pair of vertices u, v ∈ V , only
one of (u, v) and (v, u) can exist. An oriented coloring of a graph is a proper coloring that respects
the orientation. That is, if (u, v) and (x, y) are edges in the graph, then either x and v have different
colors, or, u and y have different colors. The oriented chromatic number, χo(G) of a graph with acyclic
chromatic number k is not greater than k2k−1 [RS94].
A mixed graph is a graph G = (V,A ∪ E) with vertex set V linked by arcs and edges such that the
underlying graph is simple. A is the set of ordered pair of vertices, called arcs or directed edges, and, E
is the set of unorderd pair of vertices. Directed and undirected graphs are special cases of mixed graphs.
A graph G(n,m) is a mixed graph with arcs colored with n colors and edges colored with m colors.
A colored homomorphism from a graph G1(V1, A1 ∪ E1) to a graph G2(V2, A2 ∪ E2) is any mapping
f : V1 → V2 satisfying
(x, y) ∈ A1 =⇒ (f(x), f(y)) ∈ A2, (1.1)
(x, y) ∈ E1 =⇒ (f(x), f(y)) ∈ E2, (1.2)
and, the color of the arc or edge linking f(x) and f(y) is same as the color of the arc or edge linking
x and y. The existence of a colored homomorphism from G1 to G2 is denoted by G1 → G2.
Given a graph G(V,A ∪ E), let χ(G,n,m) denote the smallest number of vertices in a graph
G′(V ′, A′ ∪ E′) such that G → G′. If the acyclic chromatic number of the underlying graph of a
G(n,m) is at most k, then χ(G,n,m) ≤ k(2n + m)k−1 [NR00].
The arboricity of an undirected graph is the minimum number of forests into which its edges can be
partitioned. Equivalently, it is the minimum number of spanning trees needed to cover all the edges of
the graph. The arboricity of a graph is a measure of how dense the graph is. Graphs with many edges
have high arboricity, and graphs with high arboricity must have a dense subgraph. The star arboricity
of G, denoted by st(G), is the minimum number of edge-disjoint star forests whose union covers all of
5
E(G). We call G, k-star-colorable if st(G) ≤ k. E(G) can be split into a(G) star forests [HMS96].
Borodin proved that every planar graph is acyclically 5-colorable [Bor06]. This along with [HMS96]
proved the conjecture by Algor and Alon [AA89] that star arboricity of a planar graph is 5.
1.5.2 Automatic Differentiation and Hessian Computation
Automatic Differentiation(AD) is a technique to evaluate the derivatives of a function defined by a
computer program. Traditional approaches like divided differences lead to truncation and approximation
errors. Automatic Differentiation tries to decompose a given vector function y = F (x) into a sequence
of elementary assignments, which are then trivially differentiated by a simple lookup table.
These elemental partial derivatives, evaluated at a particular argument, are combined in accordance
with the chain rule from derivative calculus to form some derivative information for F such as gradi-
ents, tangents, Jacobian etc. This process yields exact derivatives w.r.t. numerical accuracy. Because
the symbolic transformation occurs only at the most basic level, AD avoids the computational problems
inherent in complex symbolic computation.
A Hessian matrix, or simply the hessian is the square matrix of second-order partial derivatives of a
function; that is, it describes the local curvature of a function of many variables.
Given a real-valued function f(x1, x2, . . . , xn), if all the second partial derivatives of f exist, then
the Hessian matrix of f is the matrix
H(f)ij = DiDjf(x) (1.3)
where x = (x1, x2, . . . , xn) and Di is the differentiation operator with respect to the ith argument
and the Hessian becomes
H(f) =
∂2f∂x2
1
∂2f∂x1∂x2
· · ·∂2f
∂x1∂xn
∂2f∂x2∂x1
∂2f∂x2
2
· · ·∂2f
∂x2∂xn
......
. . ....
∂2f∂xn∂x1
∂2f∂xn∂x2
· · ·∂2f∂x2
n
(1.4)
Hessian matrices are used in large-scale optimization problems within Newton-type methods because
6
they are the coefficient of the quadratic term of a local Taylor expansion of a function.
y = f(x + ∆x) ≈ f(x) + J(x)∆x +1
2∆xTH(x)∆x (1.5)
Hessian matrices arising in large-scale applications are typically sparse. Sparsity along with symmetry
can be exploited to reduce the runtime and the storage requirement needed to compute the Hessian
using AD. Gebremedhin et al. [GTMP07] [GTPW09] have provided various methods for modelling
Hessian computation as acyclic coloring as well as algorithms for their efficient computation. Also
see [BBD+06] for a brief description.
1.5.3 Minimum Feedback Vertex Cover
A Feedback Vertex Set(FVS) of a connected graph is a subset V ′ of vertices of G such that the in-
duced graph G′ is a forest i.e. it is acyclic. The minimum feedback vertex set(MFVS) in G is an FVS
of minimum cardinality, and, is denoted by V (G). MFVS has applications in combinatorial circuit de-
sign [FLP00], distributed networks [Pel02] and optical networks [KK99].
Finding MFVS in a graph in NP-hard in general; however, it is polynomial in case of interval
graphs [LT97] and permutation graphs [Bra93] among others. [FPR09] provides a survey on the feed-
back vertex set problem.
Fertin et al. [FGR22] established the following lemmas relating the acyclic chromatic number of a
graph G and the cardinality of the MFVS in G.
Lemma 1 Let G = (V,E) be a graph of order |V | = N . If a(G) ≤ k, then |V (G)| ≤ k−2k · N .
Lemma 2 For any non-trivial graph G = (V,E), |V (G)| ≥ a(G) − 2.
They also use the following lemma established by Caragiannis et al. [CKK02].
Lemma 3 Any feedback vertex set V (G) in a graph G = (V,E) with maximum degree r satisfies
|V (G)| ≥ |E|−|V |+1r−1 .
7
Making utilization of the aforementioned lemmas and the known results on acyclic chromatic number
including those of [Gru73] [M.I79] [Bor06] [BKW99] [Sop97] [BKRS01] [BLS99], they have devel-
oped techniques to determine lower and upper bounds on the MFVS cardinality of several large families
of graphs, such as graphs of maximum degree 3, of maximum degree 4, planar graphs, planar graphs
with girth greater than equal to 5, outerplanar graphs, 1-planar graphs, k-trees etc. In some of the cases,
their bounds are tight. For all other cases, the bounds differ by a multiplicative constant of at most 2.
1.5.4 Coding Theory
Serra et al. [SZ09] have shown that acyclic coloring of a graph is equivalent to transforming the
cycle code of a graph into Maximum Distance Separable(MDS) array code. They provide description
of some families of low density MDS array codes in terms of cycle codes of colored graphs. Specifi-
cally, constructing an MDS array cycle code with distance D = 3 is same as acyclically edge coloring a
graph. The largest possible length or the number of columns is achieved for graphs with largest number
of edges i.e. a complete graph, in which case, any two colors make a Hamiltonian Path.
MDS array cycle codes of the complete graphs are called B-codes. Xu et al. [XBBW98] have shown
that B-codes have optimal encoding property with optimal lengths.
1.6 Contributions of the Thesis
The contributions of the thesis are in presenting new approaches to acyclically edge color three
classes of complete graphs which are K(p−1)(q−1),(p−1)(q−1), Kp(q−1), and Kp(q−1)(r−1).
These approaches extend the classes of graphs which can be efficiently acyclically colored in linear
time, the bounds of which are guaranteed to lie within other established results. Also, the approaches
presented are generic in nature, and, may be applied to other classes of graphs.
In addition to this, an introduction to approaches to acyclically vertex color a certain class of graphs
called grid like graphs is given. A few cases have been shown to provide an insight into possible ap-
proaches. We have also provided coloring schemes to color these graphs. Some of our schemes are
optimal while others are expected to be near optimal.
8
Chapter 2
Background and Related Work
Acyclic colorings were introduced by Grunbaum [Gru73] in 1973. Since then, it has been related
to many problems in a variety of fields. These have led to various approaches to encounter the acyclic
coloring problem (see section 1.5). In this chapter, we first review some of the common approaches that
have traditionally been used towards solving it. Later, in section 2.2, we discuss the current state of art
of this problem, and then, present some related works in the final sections.
2.1 Approaches
Among the various approaches that have been taken to tackle the problem of acyclic coloring, some
work directly on this problem while others work indirectly by aiding other approaches.
2.1.1 Greedy Coloring
Trial and error is a standard first method of approaching new probelms that helps in a better under-
standing of the problem. This generally involves a greedy approach that tries to color as many vertices
(or edges, as the case maybe) as possible. At every stage it is checked if we need to introduce a new
color. Since such an approach can take on all the possible colorings of that graph, there is a lot of back-
tracking overhead involved and thus, it has a considerable worst case complexity. One such work that
tries to greedily edge color a graph is by C.R. Subramaniam [Sub06]. His heuristic successfully edge
colors any given graph with at most 5∆(log∆ + 2) colors.
2.1.2 Examining the Structure
In this we try to exploit the underlying structure of a graph. As graphs have been divided into several
classes depending on some structural property, it is easier to analyse a structurally similar class of graphs
at the same time. Since, the structure explicitly informs us about the relations between the vertices, we
can try to color such a structural relation mathematically. A lot of work has been done in this direction by
9
focusing on parameters like the degree of a graph [Gru73] [M.I79] [Sku04] [FR08] [BC09] [SYKV09a],
or the girth of a graph [BKW99] [YHL+09]. This is also the line of approach taken by us.
2.1.3 Random Graphs
A random graph is a graph that is generated by some random process. A popular model of study is
Erdos-Renyi model which defines G(n, p) as a random graph with n vertices where each possible edge
has probability p of existing. Another closely related model is the G(n,m) model, where, of all the
possible graphs with n vertices and exactly m edges , one is randomly selected.
The study of random graphs has helped establish some bounds on certain properties of almost all
graphs. These bounds are then extrapolated to check the optimality of the deterministic approaches. A
noticeable work has been done on this line by Alon et al. [AMR91] that we will discuss later. Our work
has taken into consideration such bounds and tries to respect them to demonstrate the effectiveness of
our schemes.
2.1.4 Integer Programming Formulation
Braga et al. [BM09] have formulated the acyclic coloring problem in terms of assignment variables
and color variables. They then study different properties of the convex hull formed by the vectors of
those constraints.
2.2 State of the Art
Acyclic Colorings were introduced by Grunbaum [Gru73] in 1973 with a special emphasis on planar
graphs. He conjectured that any planar graph can be acyclically vertex colored with 5 colors. This
conjecture was later proved by Borodin [Bor06].
2.2.1 Computational Status
Determing the acyclic chromatic number is a hard problem from both a theoretical and an algorithmic
point of view. More specifically, A.V. Kostochka proved in 1978 in his thesis that it is an NP-complete
problem to decide for a given G and k if the acyclic chromatic number of G is at most k, even for
k = 3 [Kos78]. It is NP-complete even when restricted to the class of bipartite graphs [CC86]. Even
for highly structured class of complete graphs, the value of a′(G) is still not determined. Alon and Zaks
proved that it is NP-complete to determine if a′(G) ≤ 3 for an arbitrary graph G [AZ02].
10
2.3 One-factorization
1-factorization is a closely related concept which gives another way of looking at the acyclic coloring
problem. It also illustrates the difficulty of the problem. Let us formally define the concept. [Wal92]
2.3.1 Definition
A k-factor of a graph is a spanning k-regular subgraph. A k-factorization partitions the edges of a
graph into disjoint k-factors. A graph G is said to be k-factorable if it admits a k-factorization. So,
a 1-factor is a perfect matching and a 1-factorization of a k-regular graph is an edge coloring with k
colors. Note that for a graph to be k-factorable, it has to be a regular graph. In other words, a k-regular
graph is 1-factorable if it has chromatic number k.
2.3.2 Perfect 1-factorization Conjecture
Kotzig [Kot64] gave the following conjecture regarding 1-factorizations in 1963.
For any n ≥ 2, K2n can be decomposed into 2n − 1 perfect matchings such that the union of any
two matchings forms a Hamiltonian cycle.
Apart from proving that the conjecture holds for certain vaules of n, for instance, if n is prime, this
conjecture is still open [Wag92]. If such a decomposition of K2n+2 exists, then by coloring every per-
fect matching using a different color and removing one vertex we obtain an acyclic coloring of K2n+1
with 2n + 1 = ∆(K2n+1) + 1 colors. Such a coloring is best possible for K2n+1 since it is 2n-regular.
By removing another vertex from this coloring K2n+1, we obtain an acyclic edge coloring for K2n
with 2n + 1 = ∆(K2n) + 2 colors, which is best possible for K2n [AMR91]. Thus if perfect 1-
factorization conjecture is true, then a′(K2n) = a′(K2n+1) = 2n + 1 for every n.
2.4 Probabilistic Bounds
There have been many studies that establish loose upper bounds on the acyclic chromatic number.
Most of these works use the Lovasz Local Lemma. We briefly present this lemma so that the readers can
get an idea of the manner in which the results are obtained.
11
2.4.1 Lovasz Local Lemma
This lemma, proved by Laszlo Lovasz and Paul Erdos in 1975 [EL75], states that if a large number of
events are mostly independent from each other and have a low probability of occurence, then there exists
a positive probability that none of them will occur. Following is the formal statement of this lemma.
Symmetric Case: Let A1, . . . , An be events in a probability space Ω. Suppose that each event Ai is
mutually independent of a set of all the other events Aj but at most d, and that Prob[Ai] ≤ p for all i. If
ep(d + 1) ≤ 1, then Prob[∧Ai] > 0. Here e is the base of natural logarithms.
An Asymmetric version of this lemma allows the events to have different probability. Since these
works are presented only briefly, we leave this version out.
2.4.2 Alon et al.
Alon et al. [AMR91] have proved that for a graph G with maximum degree ∆, then a(G) = O(∆4/3)
as ∆ → ∞. They also show that there are graphs G with maximum degree ∆ for which a(G) =
Ω(∆4/3/(log∆)4/3).
Also in [AMR91], they have proved that any graph G with degree ∆ can be acyclically edge colored
with 64∆ colors. Molloy and Reed [MR98] improved this bound to 16∆. Muthu et al. [MNS07] have
proved a bound of a′(G) ≤ 4.52∆ for graphs G with girth at least 220.
In [ASZ01], Alon et al. have proved that there exists a constant c such that a′(G) ≤ ∆(G)+2 for any
graph G with girth at least c∆(G)log∆(G). They also conjecture that this upper bound for a′(G) holds
for all graphs G. Also, they have shown that a′(G) ≤ ∆ + 2 for almost all ∆-regular graphs. Nesetril
and Wormald have obtained a bound of a′(G) ≤ ∆+1 for a random ∆-regular (∆-fixed) graph [NW05].
Note that all these results rely heavily on probabilistic methods and are not constructive. Hence, they
may aid in guaranteeing the optimality of other deterministic or otherwise constructions.
2.5 Degree based Bounds
The degree of a graphs tells the number of neighbors of the most connected vertex. This gives an
immediate estimate of the number of colors required to color the graphs. Though, the acyclic coloring of
the graph could in principle require many more colors, degree is still an important parameter to consider
when designing a coloring scheme. It is because of this reason that the relation between the degree of a
12
graph and its acyclic chromatic number has been extensively studied compared to other approaches to
this problem.
The introductory work of Grunbaum [Gru73] started with a bound of 4 colors on acyclic vertex col-
oring of a graph with maximum degree 3. Burnstein [M.I79] showed that that any graph with maximum
degree 4 can be acyclically vertex colored with 5 colors. One can see that O(n2) colors should be suffi-
cient to acyclically vertex color a graph. Fertin and Raspaud [FR05] have given an O(n∆2) algorithm
to color any graph with∆(∆−1)
2 colors. In [FR08], they prove that 9 colors are sufficient to color a graph
with maximum degree 4.
The first linear time algorithm to acyclically edge color a subcubic graph with 5 colors was given by
Skulrattanakulchai [Sku04]. He also gave a linear time algorithm to acyclically vertex color a subcubic
graph with 4 colors.
Recently, V.Satish and K.Yadav have proved that graphs with maximum degrees 4, 5 and 6 can be
acyclically vertex colored with 5, 8 and 12 colors respectively [SYKV09a] [SYKV09b] [YSKV09] by
providing linear time algorithms that color these graphs. In [SKV09], they have proved that a graph of
maximum degree ∆ can be acyclically vertex colored with 3∆2+4∆+88 colors which is an improvement
over the result of [FR05] by a factor of 4/3.
13
Chapter 3
Coloring Schemes
In this chapter, we present linear time acyclic edge coloring schemes to color three new classes of
graphs as mentioned in Section 1.6. These graphs are a sub-class of the family of complete graphs and
complete bipartite graphs.
In [AMR91], Alon mentions that for a prime p > 2, a complete graph on p vertices can be acyclically
edge colored with p colors. Also, for a prime p > 2, a complete bipartite graph with independent sets
of sizes p − 1 each can be colored with p colors.
Since, our coloring schemes rely on these aforementioned facts, we first formally prove these before
moving on to our schemes and their proofs in the following sections. In the last section, we mention how
our schemes improve the known results for those graphs. We also discuss there, the concerns regarding
the optimality of our schemes as well as the conditions when they are expected to work well.
Before beginning, we note that a coloring scheme is said to be valid if it is proper.
3.1 Alon’s Scheme for Kp
Fact 1 A complete graph Kp with vertices 0, 1, . . . , p − 1 can be acyclically edge colored using p
colors, where p is prime greater than 2.
Scheme Color the edge (u,v) with the color (u + v) mod p.
Proof We provide a proof of both validity as well as the acyclicity of the scheme.
Validity
No two edges falling on a vertex v from any two vertices a and b can have the same color, as, (v + a)
mod p 6= (v + b) mod p, for any distinct vertices a, b.
14
Acyclicty
By Contradiction
Let there exist a bichromatic cycle of colors c1 and c2(c1 6= c2) that has length 2l, 2l ≤ p going
through vertices v1, v2, · · · , v2l, then, without loss of generality,
v1 + v2 ≡ c1 (mod p) (3.1)
and,
v2 + v3 ≡ c2 (mod p) (3.2)
=⇒
v3 ≡ (c2 − c1) + v1 (mod p) (3.3)
Similarly,
v5 ≡ (c2 − c1) + v3 (mod p) (3.4)
and so on upto
v2l+1 ≡ (c2 − c1) + v2l−1 (mod p) (3.5)
Reducing these l equations, we obtain,
v2l+1 ≡ l(c2 − c1) + v1 (mod p) (3.6)
or, as v2l+1 is v1 itself,
v1 ≡ l(c2 − c1) + v1 (mod p) (3.7)
=⇒
l ≡ 0 (mod p) (3.8)
or,
l = kp for some k ≥ 1 (3.9)
But as, l ≤ p/2, this is a contradiction. Hence there does not exist any bichromatic cycle with edge-
coloring scheme.
3.2 Alon’s Scheme for Kp−1,p−1
Fact 2 A complete bipartite graph Kp−1,p−1 with vertices 0, 1, . . . , p − 2 in each of the independent
sets, can be acyclically edge colored using p colors, where p is prime greater than 2.
15
Scheme Color edge (u,v) with the color (u + v) mod p.
Proof As before, we provide a proof of both properness as well as the acyclicity of the scheme.
Validity
No two edges falling on a vertex v from any two vertices a and b on the opposite partition can have the
same color, as, (v + a) mod p 6= (v + b) mod p, for any distinct vertices a, b.
Acyclicity
By Contradiction
Let there exist a bichromatic cycle of colors c1 and c2(c1 6= c2) of length 2l, 2l ≤ 2(p − 1) going
through vertices v1, v2, · · · , v2l. Note that these vertices alternate from one independent set to the other.
Proceeding in the same manner as before,
v1 + v2 ≡ c1 (mod p) (3.10)
and,
v2 + v3 ≡ c2 (mod p) (3.11)
=⇒
v3 ≡ (c2 − c1) + v1 (mod p) (3.12)
Similarly,
v5 ≡ (c2 − c1) + v3 (mod p) (3.13)
and so on upto
v2l+1 ≡ (c2 − c1) + v2l−1 (mod p) (3.14)
Reducing these l equations, we obtain,
v2l+1 ≡ l(c2 − c1) + v1 (mod p) (3.15)
or, as v2l+1 is v1 itself,
v1 ≡ l(c2 − c1) + v1 (mod p) (3.16)
=⇒
l ≡ 0 (mod p) (3.17)
or,
l = kp for some k ≥ 1 (3.18)
But as, l ≤ (p − 1), this is a contradiction. Hence there does not exist any bichromatic cycle with
this edge-coloring scheme.
16
3.3 Complete bipartite graph K(p−1)(q−1),(p−1)(q−1)
In this section, we present our coloring scheme for acyclically edge coloring a complete bipartite
graph with independent sets of size (p − 1)(q − 1) vertices each, K(p−1)(q−1),(p−1)(q−1) where p, q are
primes. Here, we make use of the intution that a K(p−1)(q−1),(p−1)(q−1) can be viewed as a set of several
complete bipartite graphs. More specifically, as Figure 3.1 shows, we can view a K(p−1)(q−1),(p−1)(q−1)
as a graph with multi-nodes and multi-edges. Each of these (p − 1) multi-nodes on either side is a set
of (q − 1) vertices, and each of the multi-edges is a set (q − 1)2 edges. This view makes this new graph
completely bipartite on independent sets with (p − 1) multi-nodes each.
multi−nodesmulti−edges
Figure 3.1 A complete bipartite graph K8,8, viewed as K(3−1)(5−1),(3−1)(5−1)
This technique allows us to use the coloring scheme of Kp−1,p−1 on this new graph successively to
color the original graph, i.e. K(p−1)(q−1),(p−1)(q−1).
Claim 1 We can acyclically edge-color K(p−1)(q−1),(p−1)(q−1) with pq colors, where p, q are primes
greater than 2.
Scheme
Step 1: For each of the independent sets of the bipartite graph, partition the (p−1)(q−1) vertices
into (p − 1) sets of size (q − 1) each. That is, if the vertices in each partition are numbered as
0, 1, 2, . . . , (p−1)(q−1)−1, partition them as Vi = i(q−1), i(q−1)+1, . . . , i(q−1)+(q−2),
for i = 0, 1, 2, . . . , p − 2.
Step 2: For every i, j ∈ 0, 1, 2 . . . p− 2, and for every pair of vertices, say u ∈ Vi and v ∈ Vj ,
color the edge (u, v) with the color ((i + j) mod p, (u + v) mod q).
17
Proof
Clearly, this algorithm uses pq colors in total as the first and second color coordinates can be upto p
and q respectively. Also, since there are edges that use the color (p−1, q−1), this bound on the number
of colors is tight as well.
We rename a vertex u ∈ Vi as (ua, ub) such that a = i and b = u−(q−1)i, such that 0 ≤ a < p−1,
0 ≤ b < q − 1.
Validity
No two edges falling on a vertex (vi, vj) in Kp(q−1) from any two vertices (va, vb) and (vc, vd) can have
same color. Only two cases arise here.
1. a = c, i.e. (va, vb) and (vc, vd) are in the same partition. Since these are two different vertices,
b 6= d. So, (j + b) mod q 6= (j + d) mod q. Hence these two edges will have different colors.
2. a 6= c i.e. (va, vb) and (vc, vd) are in the different partitions. So, (i + a) mod p 6= (i + c) mod p.
Thus, the colors of the edges differ.
Hence, the coloring scheme is valid.
Acyclicity
By Contradiction
Let there exist a bichromatic cycle with colors (c1, c′1), (c2, c
′2) of length 2l, 2l ≤ 2(p− 1)(q − 1) going
through vertices (v1, w1), (v2, w2), . . . , (v2l, w2l), then,
v2 + v1 ≡ c1 (mod p) (3.19)
v3 + v2 ≡ c2 (mod p) (3.20)
From (3.19) and (3.20), we obtain,
v3 ≡ (c2 − c1) + v1 (mod p) (3.21)
Similarly,
v5 ≡ (c2 − c1) + v3 (mod p) (3.22)
or
v5 ≡ 2(c2 − c1) + v1 (mod p) (3.23)
18
and reducing subsequent equations so on, we get,
v2l+1 ≡ l(c2 − c1) + v1 (mod p) (3.24)
or
v1 ≡ l(c2 − c1) + v1 (mod p) (3.25)
Proceeding in the same manner for colors c′1 and c′2,
w2 + w1 ≡ c′1 (mod p) (3.26)
w3 + w2 ≡ c′2 (mod p) (3.27)
From (3.26) and (3.27), we obtain,
v3 ≡ (c2 − c1) + v1 (mod p) (3.28)
Similarly, we can have equations corresponding to equations (3.22) through (3.24).
Using these aforesaid equations, we arrive at the following equation similar to equation (3.25)
w1 ≡ l(c′2 − c′1) + w1 (mod q) (3.29)
Now, since the coloring is proper, at least one of the following must hold true.
c1 6= c2 (3.30)
c′1 6= c′2 (3.31)
Three cases arise here:
Case 1: Both of the equations (3.30) and (3.31) hold true. Then, equations (3.25) and (3.29) maybe
rewritten as
l ≡ 0 (modp) (3.32)
and,
l ≡ 0 (modq) (3.33)
respectively.
19
Equations (3.32) and (3.33) taken together imply that
l = kpq for some k ≥ 1 (3.34)
This is not possible as l ≤ (p − 1)(q − 1), whereas equation (9) implies that l ≥ pq. This leads
to a contradiction.
Case 2: Equation (3.30) is true and (3.31) is false. So c′1 = c′2. In this case, equation (3.25) maybe
rewritten as
l ≡ 0 (modp) (3.35)
Now,
w1 + w2 ≡ c′1 ≡ c′2 ≡ w2 + w3 (mod q) (3.36)
=⇒
w1 ≡ w3 (mod q) (3.37)
Proceeding in the same manner, we obtain,
w1 ≡ w3 ≡ w5 ≡ · · · ≡ w2l−1 (mod q) (3.38)
and,
w2 ≡ w4 ≡ w6 ≡ · · · ≡ w2l (mod q) (3.39)
But, for any x, y ∈ w1, w3, . . . , w2l−1, there do not exist two distinct x, y < (q − 1) in the
same partition for equation (3.38) to hold true. The same holds for any two vertices from the set
w2, w4, . . . , w2l.
This bring us to the first of our three important observations regarding this proof.
Observation 1: Alternate vertices belong to distinct partitions.
Secondly, we note that if any vertex in the cycle has second coordinate as wa, then its neighboring
vertices in the cycle will have second coordinate of the form wb such that
a + b ≡ c′ (mod q) ; where c′ = c′1 = c′2. (3.40)
Since, for a given a < q − 1, there exists a unique b < q − 1 such that equation (3.40) holds,
and vice versa; This means that any vertex in the cycle will have second coordinate of the form of
20
either wa or wb. This can be summarised in our next observation.
Observation 2: Any partition can contribute at most one vertex to the cycle.
Note that the first coordinates of the cycle alternate; any color of the form (X + X) mod p
can appear at most twice as the two colors that alternate. This is true because, for any disctinct
X,Y < (p − 1), (X + X) mod p 6= (Y + Y ) mod p. Hence, we state our next observation.
Observation 3: If a partition VA, 0 ≤ A < p − 1, from one of the independent sets appears in
the cycle, then the partition VA in the other independent set cannot occur in the cycle, except for
at most twice.
Observations 1, 2 and 3 take together imply that any bichromatic cycle where c′1 = c′2 can be of
length at most p + 1. This implies l ≤ p+12 . But, this is a contradiction to equation (3.35).
Case 3: Equation (3.31) is true and (3.30) is false. So, c1 = c2. In this case, equation (3.29) maybe
rewritten as
l ≡ 0 (mod q) (3.41)
Suppose the cycle passed through a partition Vi in the one of the independent sets, and, the par-
titions preceeding and succedding it in the other independent set are Vj and Vk respectively, for
some i, j, k ∈ 0, . . . , p − 2. Since the first color coordinates are same,
i + j ≡ i + k (mod p) (3.42)
=⇒
j ≡ k (mod p) (3.43)
But, since j, k < p − 1, this implies j = k. Hence such a cycle will involve only two partitions.
Thus, 2l ≤ 2(q − 1), or, l ≤ q − 1. This is a contradiction to equation (3.41).
Hence, our scheme is a valid acyclic edge coloring scheme for K(p−1)(q−1),(p−1)(q−1).
21
3.4 Complete graph Kp(q−1)
In this section, we again use the concept of multi-nodes and multi-edges to present a coloring scheme
to acyclically edge coloring a complete graph on p(q − 1) vertices with p, q being prime. This time,
we try to envision a Kp(q−1) as a set of complete bipartite graphs that are completely connected. More
specifically, as the Figure 3.2 shows, each of the p multi-nodes is a set of q− 1 vertices, and, each of the
multi-edges is a set of (q − 1)2 edge. This makes this new graph a complete graph on p multi-nodes.
Each of the multi-edges resembles a complete bipartite graph except for the fact that the set of vertices
within a multi-node do not form an independent set. Rather, they are all completely connected!
multi−edge
multi−node
Figure 3.2 A complete graph on 10 vertices, K10, viewed as K5(3−1)
This technique allows us to use the coloring scheme for Kp on this new graph and the coloring
scheme for Kp−1,p−1 on each of the multi-edges. Of course, this needs to be adjusted to work overall
for the original graph, i.e. Kp(q−1).
Claim 2 We can acyclically edge-color Kp(q−1) with (2p − 1)q colors, where p, q are primes greater
than 2.
Scheme
Step 1: Partition the p(q − 1) vertices into p sets of size (q − 1) each. That is, if the vertices
are numbered as 0, 1, 2, . . . , p(q − 1) − 1, partition the vertices as Vi = i(q − 1), i(q − 1) +
1, . . . , i(q − 1) + (q − 2), for i = 0, 1, 2, . . . , p − 1.
Step 2: For every i, j(i 6= j) ∈ 0, 1, 2, . . . , p − 1, and for every pair of vertices, say, u ∈ Vi
22
and v ∈ Vj , color the edge (u, v) with the color ((i + j) mod p, (u + v) mod q).
Step 3: For every i ∈ 0, 1, 2, . . . , p− 2, and for every pair of vertices, say, u, v ∈ Vi , color the
edge (u, v) with the color ((p + i), (u + v) mod q).
Step 4: For i = p − 1, and for every pair of vertices, say, u, v ∈ Vi , color the edge (u, v) with
the color ((i + i) mod p, (u + v) mod q), or (p − 2, (u + v) mod q).
Proof
First, we note that this algorithm uses (2p − 1)q colors in total as the first and second color coordi-
nates can be upto 2p−1 and q respectively. Also, since there are edges that use the color (2p−2, q−1),
this bound on the number of colors is tight as well.
As with before, we rename a vertex u ∈ Vi as (ua, ub) such that a = i, and, b = u − (q − 1)i, such
that 0 ≤ a < p, 0 ≤ b < (q − 1).
Validity
No two edges falling on a vertex (vi, vj) in Kp(q−1) from any two vertices (va, vb) and (vc, vd) can have
same color. Three cases arise here.
1. a = c, i.e. (va, vb) and (vc, vd) are in the same partition. Since these are two different vertices,
b 6= d. So, (j + b) mod q 6= (j + d) mod q. Hence these two edges will have different colors.
2. a 6= c i.e. (va, vb) and (vc, vd) are in the different partitions, and, either i = a or i = b. In
this case, the edge within the same partition has a first color coordinate greater than or equal to p
whereas the other edge has first color coordinate less than p.
3. a 6= c, and, i 6= a, and, i 6= b, i.e. all three vertices lie in different partitions. In this case,
(i + a) mod p 6= (i + c) mod p.
Thus, the colors of the edges differ.
Hence, the coloring scheme is valid.
Acyclicity
By Contradiction
We begin by assuming the existence of a bichromatic cycle with our coloring scheme. We then take up
different cases and prove that in each case, such an assumption leads to a contradiction.
23
External Cycles The cycle does not involve any of the edges within a partition.
Let there exist a bichromatic cycle with colors (c1, c′1), (c2, c
′2) of length 2l, 2l ≤ p(q − 1) going
through vertices (v1, w1), (v2, w2), . . . , (v2l, w2l). In this case, we can obtain equations similar to
equation (3.19) through (3.29) used in the proof in the previous section. Hence, we arrive at
v1 ≡ l(c2 − c1) + v1 (mod p) (3.44)
and
w1 ≡ l(c′2 − c′1) + w1 (mod q) (3.45)
Also, for the coloring to be proper, at least one of the following must hold true:
c1 6= c2 (3.46)
c′1 6= c′2 (3.47)
Three cases arise here:
Case 1: Both of the equations (3.46) and (3.47) hold true. Then, equations (3.44) and (3.45)
maybe rewritten as
l ≡ 0 (modp) (3.48)
and,
l ≡ 0 (modq) (3.49)
respectively.
Equations (3.48) and (3.49) taken together imply that
l = kpq for some k ≥ 1 (3.50)
This is not possible as l ≤ p(q−1)2 , whereas equation (3.47) implies that l ≥ pq. This leads
to a contradiction.
Case 2: Equation (3.46) is true and (3.47) is false. So, c′1 = c′2. In this case, equation (3.45)
maybe rewritten as
l ≡ 0 (modp) (3.51)
24
We will make two observations here.
If any vertex in the cycle has second coordinate as ua, then its neighboring vertices in the
cycle will have second coordinate of the form ub such that
a + b ≡ c′ (mod q) ; where c′ = c′1 = c′2. (3.52)
Since, for a given a < q − 1, there exist a unique b < q − 1 such that equation (3.52) holds,
and vice versa; every vertex in the cycle will have second coordinate of the form of either
ua or ub. Since, every partition has only one vertex with second color coordinate as ub, only
two vertices can be part of the cycle at most. Hence,
Observation 1: Any partition can contribute at most two vertices to the cycle.
Since, edges within a partition cannot be used, the cycle cannot utilise both of the available
vertices from each of the partitions. This gives our second observation.
Observation 2: At least one partition cannot contribute two vertices to the cycle.
Observations 1 and 2 taken together imply that any such bichromatic cycles will have a
length less than 2p, or l < p. This leads to a contradiction with equation (3.51).
Case 3: This case is exactly similar to the corresponding case in the proof of Claim 1 in previous
section.
Therefore, there cannot exist any bichromatic cycle that uses external edges alone.
Internal Cycles The bichromatic cycle can involve edges within a partition.
With an exception of partition p − 1, the edges within all the partitions 0 through p − 2 have a
distinct first color coordinate from any of the other edges in the entire graph. Hence, the only
possible bichromatic cycles can occur within these partitions. Since, the second color coordinates
within a partition have the same colors as assigned to a complete graph on q vertices where q is
a prime, there cannot be any bichromatic cycles within a partition as well. Refer to the proof of
25
Fact 1 in section 3.1.
As far as partition p − 1 is concerned, we note the following.
1. No bichromatic cycles can occur within itself because of the reason noted above.
2. Any possible cycle that uses an edge outside of the partition, can have at most one edge from
the partition because any the first color coordinate of the any edge coming from outside of the
partition to itself is different from the first color coordinate of the edges within this partition.
3. Both of the edges that precede and succeed the involved edge of this partition must come from
the same partition as they have the same color. Since, none of the edges further from these two
connecting edges can have the same color as that of the involved edge from this partition, the
cycle cannot be completed.
Hence, there cannot exist any bichromatic cycle which involves any internal edge.
Therefore, we state that the scheme presented is a valid acyclic edge coloring scheme for Kp(q−1).
3.5 Complete graph Kp(q−1)(r−1)
In this section, we present a coloring scheme to acyclically edge color a complete graph on p(q −
1)(r − 1) vertices, Kp(q−1)(r−1) where p, q, r are primes. As with the previous schemes, we again rep-
resent this graph as a graph on multi-nodes and multi-edges. In this case, we represent a Kp(q−1)(r−1)
as a complete graph on p multi-nodes, with each of the multi-edges representing the (p − 1)(q − 1)2
edges of the complete bipartite graph K(p−1)(q−1),(p−1)(q−1). Figure 3.3 shows an example of a graph
with five multi-nodes. The edges as encapsulated in the multi-edges shown.
Now with this encapsulation, we can use the coloring schemes of earlier sections to build a coloring
scheme to acyclically edge color this graph keeping in mind the structural constraints imposed by it on
the possible schemes.
Claim 3 We can acyclically edge-color Kp(q−1)(r−1) with (2p − 1)(2q − 1)r colors, where p,q,r are
primes greater than 2.
Scheme
Step 1: Partition the p(q − 1)(r − 1) vertices into p sets of size (q − 1)(r − 1) each. That
is, if the vertices are numbered as 0, 1, 2, . . . , p(q − 1)(r − 1) − 1, partition the vertices as
26
multi−edges
multi−nodes
Figure 3.3 A complete graph on 5 multi-nodes of Kpq for some primes p, q greater than 2
Vi = i(q − 1)(r − 1), i(q − 1)(r − 1) + 1, . . . , i(q − 1)(r − 1) + (q − 1)(r − 1) − 1, for
i = 0, 1, 2, . . . , p − 1.
Step 2: For every pair of vertices, say u ∈ Vi and v ∈ Vj , where i 6= j, assign the first color
coordinate of the edge (u, v) as ((i+ j) mod p). If, i = j, assign the first color coordinate as p+ i
for 0 ≤ i < p − 1, and as p − 2 when i = p − 1.
Step 3: Each of the multi-nodes is colored with the coloring scheme for a Kq(r−1) from previous
section, and each of multi-edges is given color following the coloring of a K(q−1)(r−1),(q−1)(r−1)
from section 3.3. These colors form the second and third coordinate of the color assigned to the
edges.
Proof
Firstly, this algorithm can be seen to use (2p − 1)(2q − 1)r colors in total. Also, this bound is tight
as well.
We use similar naming convention as before to help us with the proof. We rename a vertex u ∈ Vi as
(ua, ub, uc) such that a = i. Let k = u− i[(q− 1)(r− 1)], then, b = k/(r− 1) and c = k mod (r− 1),
such that 0 ≤ a < p, 0 ≤ b < (q − 1), 0 ≤ c < (r − 1).
For this proof, we will frequently rely on the results already proved in the previous sections. This
also clearly brings out the manner in which the aforementioned schemes have been juxtaposed to form
a working scheme for Kp(q−1)(r−1).
Validity
For this coloring to be valid, no two edges incident on any vertex v should have the same color. Here
27
we mention three possible cases.
1. If both edges belong to the same multi-node. Clearly they cannot have the same color by proof of
validity of Claim 2 in previous section.
2. If both edges belong to the same multi-edge. Then, they cannot have the same color by proof of
validity of Claim 1 in previous section 3.3.
3. If the edges do not fall in either of the above mentioned cases, then they would have different first
color coordinate.
Hence, the coloring scheme is valid.
Acyclicity
By Contradiction
We assume that there exists a bichromatic cycle in the graph after it has been colored with our coloring
scheme. As with proof of Claim 2, we consider different setting when such a cycle can occur and prove
that it leads to a contradiction each time.
External Cycles The cycle does not involve any edge within a partition.
Let there exist a bichromatic cycle with colors (c1, c′1, c
′′1), (c2, c
′2, c
′′2) of length 2l, 2l ≤ p(q −
1)(r − 1) going through vertices (v1, w1, x1), (v2, w2, x2), . . . , (v2l, w2l, x2l). We obtain equa-
tions similar to equations (3.44) and (3.45) used in the proof of Claim 2 in last section.
v1 ≡ l(c2 − c1) + v1 (mod p) (3.53)
w1 ≡ l(c′2 − c′1) + w1 (mod q) (3.54)
and
x1 ≡ l(c′′2 − c′′1) + x1 (mod r) (3.55)
Now, since the coloring is known to be valid, at least one of the following must hold true.
c1 6= c2 (3.56)
28
c′1 6= c′2 (3.57)
c′′1 6= c′′2 (3.58)
Now, we consider the possible manners in which the last statement holds.
Case 1: All three of the above equations hold true. Then they maybe rewritten as
l ≡ 0 (mod p) (3.59)
l ≡ 0 (mod q) (3.60)
and
l ≡ 0 (mod r) (3.61)
Equations (3.59), (3.60) and (3.61) taken together imply that
l = kpqr for some k ≥ 1 (3.62)
But since, l ≤ p(q−1)(r−1)2 , this is not possible. Hence, it is a contradiction.
Case 2: Equation (3.56) is false, i.e. c1 = c2. This reduces to the proof to acyclicity of Claim 1
of section 3.3. Hence, a bichromatic cycle cannot exist in this case as well.
Case 3: Each of the three cases where either or both of equations (3.57) and (3.58) are false, get
reduced to the corresponding cases of proof of Claim 2 in the last section.
Hence, we state that there does not exist any bichromatic cycle which does not use any edge from
within a partition.
Internal Cycles Clearly, for all the partitions there cannot exist any bichromatic cycle which does in-
clude any edge from outside the partition since they use the coloring scheme of Kq(r−1) which is
shown to be acyclic in the previous section. As for the cycles which do involve the edges from
outside the partition, we note that the first color coordinates of edges of each of the partitions
from 0 through p − 2 are distinct from each other as well as from any of the other edges in the
entire graph. So, they cannot contribute to any bichromatic cycle. For the case of partition p − 1,
we mention that the exact same arguments as presented in the corresponding part in the proof of
Claim 2 fit in here to ensure that there cannot exist any bichromatic cycles involving any internal
edge of partition p − 1.
29
Therefore, we state that the scheme presented here is a valid coloring scheme that acyclically edge
colors the graph Kp(q−1)(r−1).
3.6 Extension and Optimality
We note that the schemes provided build upon themselves and adapt properly to acyclically edge
color bigger classes of graphs. One may be tempted to ask if the scheme can be extended to graph on
number on vertices which could be a product of any number primes or primes-1.
A straight answer is that the schemes are extremely flexible with respect to the ability to be adapted
to work upto very large number of primes or primes-1 as factors to the number of vertices. But, as we
know from Fact 1 and Fact 2 that any complete graph on n vertices can be acyclically edge colored with
a number of colors equal to the next prime greater than or equal to n. Similarly, any complete bipartite
graph with independent sets of size n each can be acyclically edge colored with a number of colors
equal to the next prime greater than n. Although such a scheme would not be optimal with respect to
the number of colors used, it provides an upper bound for the acyclic chromatic number. Hence, any
efficient (w.r.t. number of colors used) coloring scheme that we propose, should be able to limit the
number of colors used so as to not exceed the next prime. If this condition is violated, then even though
our schemes would be valid acyclical edge coloring schemes they would not be the optimal schemes.
Since prime gaps are known to increase extremely fast as the size of the primes increase, our schemes
are expected to perform well for most cases. Also, it is known that there is always a prime number be-
tween n and 2n for all n > 3. Hence, for our coloring scheme of Claim 2 can be expected to work
optimally when (2p − 1)q < 2p(q − 1) or q > 2p. Similar formulation for Claim 3 to be optimal in
number of colors would lead to r(2pq + 1) < 2[q(p + r) + p]. The scheme used in Claim 1 uses pq
colors to acyclically edge color 2(p − 1)(q − 1) vertices of a complete bipartite graph. This is expected
to work well mostly.
30
Chapter 4
Grid like Graphs
In this chapter we study a class of graphs referred to as d-dimensional partial tori. These are graphs
that can be expressed as the cartesian product of d graphs. This class includes some known classes of
graphs like d-dimensional meshes, hypercubes, tori etc. In the first two sections, we define the graph
product operation and formally introduce the d-dimensional partial tori graphs. Then, we mention the
state of art of the colorings for these graphs. In the final section, we present some extensions to the work
done. We give efficient linear time acyclic vertex coloring schemes for some basic graphs of the family
of d-dimensional partial tori. Some of these are optimal while others are close to optimal.
4.1 d-dimensional Partial Tori
We use Pk to denote a simple path on k vertices such that V (Pk) = 0, . . . , k − 1 and E(Pk) =
(i, j) : |i − j| = 1. Similarly, Ck is used to denote a cycle on k vertices such that V (Ck) =
0, . . . , k − 1 and E(Ck) = E(Pk) ∪ (k − 1, 0). We use PATHS to denote the set P2, P3, . . . of all
paths on 2 or more vertices. Similarly, CYCLES is used to denote the set C3, C4, . . . of all cycles.
A d-dimensional partial torus is a connected graph G whose unique prime factorization is of the form
G = G12 · · ·Gd where Gi ∈ PATHS ∪ CYCLES for each 1 ≤ i ≤ d. Here the operator 2 represents
graph product which is defined as following.
4.2 Graph Product
Given two graphs G1 = (V1, E1) and G2 = (V2, E2), the cartesian product of G1 and G2, denoted
by G12G2, is defined to be the graph G = (V,E) where V = V1 × V2 and E contains the edge joining
(u1, u2) and (v1, v2) if and only if either u1 = v1 and (u2, v2) ∈ E2 or u2 = v2 and (u1, v1) ∈ E1.
31
Note that the graph product operation is commutative i.e. G12G2 and G22G1 are isomorphic.
They are similarly associative. Hence, the graph G12 · · ·2Gd is unambiguously defined for any d.
Gd denotes the d-fold cartesian product of a graph with itself. Sabisussi [Sab59] and Vizing [Viz63]
have shown that any connected graph G can be expressed as a product G12 · · ·2Gk of primes factors
Gi(1 ≤ i ≤ k). A graph is said to be prime with respect to the 2 operation if it has at least two vertices
and it is not isomorphic to the product of two non-trivial graphs (having at least two vertices). Also,
this factorization is unique except for a re-ordering of the factors and is referred to as the Unique Prime
Factorization of the graph. Since a(G) is a graph invariant, we assume without loss of generality that
any graph that is either an invariant path or an induced cycle has all the factors from PATHS∪CYCLES.
Notice that when for a graph G, each of the factors Gi = P2, then G is a d-dimensional hyper-
cube. Similarly, when each of the Gi ∈PATHS, then G is a d-dimensional mesh, and, when every
Gi ∈CYCLES, then G is a d-dimensional torus.
Let G1 and G2 be two families of graphs. Then, we define the operation 2 between G1 and G2 such
that
G12G2 = G12G2|G1 ∈ G1 and G2 ∈ G2
4.3 Acyclic Edge Coloring
Muthu et al. [MNS06] have provided polynomial time constructive schemes to acyclically edge color
the class of d-dimensional partial tori. Their estimates are exact except for one case that we shall de-
scribe later. Follwing are their results for the special cases when each of the factor graphs belongs to
only one of either P2, PATHS or CYCLES.
a′(P d2 ) = ∆(P d
2 ) = d + 1 if d ≥ 2; a′(P2) = 1. (4.1)
a′(G) = ∆(G) = 2d for each G ∈ Md (4.2)
a′(G) = ∆(G) + 1 = 2d + 1 for each G ∈ Td (4.3)
They also proved the general results when any given graph is taken product with either a P2, a
member of PATHS, or a member of CYCLES. If a′(G) = η
a′(G2P2) ≤ η + 1, if η ≥ 2. (4.4)
a′(G2Pl) ≤ η + 2, if η ≥ 2 and l ≥ 3. (4.5)
32
a′(G2Cl) ≤ η + 2, if η ≥ 2 and l ≥ 3. (4.6)
They have posted the acyclic edge chromatic number of several variants of d-dimensional partial tori
when it is a product of certain number of members of P2 or PATHS or CYCLES etc.
4.4 Acyclic Vertex Coloring
Fertin et al. [FGR03] have explored the acyclic vertex coloring of grids which are a subclass of the
more general d-dimensional partial tori graphs. In other words, none of the prime factors of the graphs
that they consider belongs to CYCLES. They describe lower and upper bounds for acyclic chromatic
number of d-dimensional grid G(n1, . . . , nd). G(n1, . . . , nd) denotes that G has ni vertices in the ith
dimension (1 ≤ i ≤ d). They also show that these bounds match, and thus give an optimal result, when
the length in each dimension is sufficiently large, or more precisely, if∑d
i=11ni
≤ 1. If that is not the
case, then these bounds differ by an additive constant of at most |1−⌊∑d
i=11ni
⌋|. Also, they apply those
results to a hypercube of d dimensions, which is a special case of G(n1, . . . , nd) in which there are only
2 vertices in each dimension. In this case, their bounds differ by a multiplicative constant of 2.
4.5 Colorings
We try to acyclically vertex color the class of d-dimensional partial tori graphs that involve at least
one member from CYCLES as its prime factor. We try to use techniques to exploit the structure of the
d-dimensional partial tori graphs. In this respect, we begin forward on a case by case basis as done in
the work of Muthu et al. [MNS06]. We provide some cases where our anaysis works. We also provide
a corresponding acyclic vertex coloring scheme for the given graph.
4.5.1 CYCLES 2 PATHS
In this subsection we consider the graphs where a member of CYCLES is taken product with a mem-
ber of PATHS. Notice that this leads to forming multiple copies of the given cycle that are connected
with corresponding vertices in the copies. The number of such copies being equal to the length of the
path. This gives us the advantage of symmetry in the manner that any scheme that works within a cycle
works for all the copies. So, we focus only on the edges connecting them. Each of the following cases
describes this further.
33
4.5.1.1 Cm2P2,m is odd
We consider the coloring of the graphs produced when a cycle of odd length Cm (m is odd), is taken
product with a P2. Figure 4.1 shows this resulting graph. As observed before, this looks like a graphs
with two cycles of same length connected at the corresponding vertices. So, we can hope to modify the
acyclic coloring of the cycle and apply it to the copy resulting in the acyclic vertex coloring of the whole
graph.
3
2
3
3
2
1
1
2
1
2
Figure 4.1 A Cm2P2, where m is odd
Lemma 1. Cm2P2 can be acyclically vertex colored with 3 colors for all oddm.
Proof. First of all, we note that a(Cm) = 3 for all m > 2. So, let us assume without loss of general-
ity, that the operand Cm was colored with 3 colors as shown in Figure 4.1. Clearly this given scheme
acyclically vertex colors Cm. Now, let us denote the copy as C ′m. We propose to modify this scheme to
suitably color C ′m as follows.
If v is a vertex in the operand cycle, then let σ(v) denote the corresponding vertex in C ′m. Then, if
c(v) denotes the color of v, then
c(σ(v)) = c(v) mod 3 + 1 (4.7)
Now, we see that such a vertex coloring is proper as well as acyclic for the product graph. Clearly,
this coloring is proper since there are no adjacent vertices with the same color. This is because
34
1. Cm is given to be properly colored. So, C ′m is properly colored as well.
2. c(σ(v)) 6= c(v) for any v.
To show that this coloring is acyclic we consider all the possible bichromatic cycles that can occur.
Also note that since both Cm and C ′m are acyclically colored, any such cycle should involve vertices
from both of them.
1. 1 − 2 · · · − 1 cycle This cannot happen since the only 1-colored vertex in C ′m does not have two
neighbors with color 2.
2. 2 − 3 · · · − 2 cycle The only 3-colored vertex in Cm does not have two neighbors with color 2.
So, this cannot occur as well.
3. 3 − 1 · · · − 3 cycle None of the 1-colored vertices in Cm have two 3-colored neighbors.
Hence, this graph can be colored with 3 colors which is also optimal. Thus, the acyclic chromatic
number of the graph, Cm2P2 is 3 when m is odd.
4.5.1.2 Cm2P2,m is even
Lemma 2. The acyclic chromatic number of Cm2P2 is 3 i.e. a(Cm2P2) = 3 for allm 6= 4.
Proof. In this case we assume that the given Cm is colored with 3 colors as shown in Figure 4.2. Such
a coloring is clearly acyclic. Notice how the last four vertices of the cycles have been colored to make
it look similar to the last case to reuse the conditions that prevent a bichromatic cycle. We also note that
this requires m to be at least 6. Now we try to color the product graph based on this.
As before, we denote the copy of the operand Cm as C ′m and we define a function σ that maps the
colors from the operand Cm to C ′m. In fact, we use the same σ as before i.e.
c(σ(v)) = c(v) mod 3 + 1 (4.8)
Clearly, this coloring is proper by the same argument as before. To show that it is also acyclic, we
again go through all possible bichromatic cycles. We also mention here that since both the Cm as well
as C ′m are acyclically colored, any such cycle must involve both of them.
1. 1 − 2 · · · − 1 cycle This cannot happen since the only 1-colored vertex in C ′m does not have two
neighbors with color 2.
2. 2− 3 · · · − 2 cycle The only 3-colored vertices in Cm do not have two neighbors with color 2. So,
this cannot occur as well.
35
2
1 2
3
1
2
3
1
2
3
1
2
2
3
3
1
extra edge
Figure 4.2 A Cm2P2, where m is even
3. 3 − 1 · · · − 3 cycle None of the 3-colored vertices in C ′m have two neighbors that are 1-colored.
Hence, this coloring scheme optimally colors the given graph with 3 colors. By clubbing this result
with the previous one, we can state that the acyclic chromatic number of Cm2P2 is 3 for all m 6= 4.
4.5.1.3 Cm2Pk
Now we extend our previous results to the case of Cm2Pk (k>2), i.e. we now have more than just
two copies of the operand Cm. Notice that we can color alternate copies of Cm with same color i.e.
all odd copies with one color scheme and all even copies with another such that they do not share any
color, then such a scheme would acyclically color the given graph. Such a scheme would require 6 col-
ors. Figure 4.3 shows an example with odd copies using the colors 1, 2, 3 and the even copies using
colors 4, 5, 6.
The reason such a scheme works is because
1. No color pair from one copy can be extended to copies adjacent to it as the adjacent copies do not
use those colors. Also, it is given that none of Cm can form a bichromatic cycle within themselves.
2. Any color pair that use one color each from two consecutive copies cannot complete a bichromatic
cycle, because, such a path cannot come back to complete the cycle. In other words, any path that
leads back to the first vertex of such a bichromatic cycle will have to use a vertex other than these
two colors.
36
3
1
2
1
2
1
2
1
2
3
4
5
4
5
6
Figure 4.3 A Cm2P3 colored with 6 colors, where m is odd
So, a(Cm2Pk) ≤ 6. We now try to reduce any colors that can be dispensed with.
Lemma 3. Cm2Pk can be acyclically vertex colored with 4 colors for allm and k > 2.
Proof. Note that the only color pair in the given Cm that forms a bichromatic path is 1, 2. So, we just
need to avoid one of these colors in any adjacent copy and the rest of the colors can be reused. Figure
4.4 shows such a coloring.
3
1
2
1
2
1
2
1
2
3
4
4
3
3
1
Figure 4.4 A Cm2P3 colored with 4 colors, where m is odd
37
We formally define this coloring scheme by providing a mapping function from the colors in the
operand Cm to corresponding vertices in the even copies. As before, we define such a function that
maps the colors of operand graph Cm to the copies as following.
c(σ(v)) = 4 if c(v) = 1 (4.9)
= 3 if c(v) = 2 (4.10)
= 1 if c(v) = 3 (4.11)
Notice that this is same as the function before except for that a new color 4 has been introduced to
replace color 2 in the mapping for the above mentioned reason. Clearly, this coloring is acyclic. This
can be proved with the same arguments as presented in the initial scheme.
We note that the case where m is even can also be colored in a similar manner. Hence, Cm2Pk can
be colored with 4 colors for all m and k (m,k > 2).
4.5.1.4 Optimality of coloring of Cm2Pk
In 4.5.1.3, we proved that a Cm2Pk can be acyclically vertex colored with 4 colors. In this section,
we prove that this is also the optimal number of colors required to acyclically vertex color it.
Lemma 4. Acyclic chromatic number of Cm2Pk is 4 i.e. a(Cm2Pk) = 4.
Proof. As noted before, we know that 4 colors are sufficient to color a Cm2Pk. Now, we show that
there exists atleast one family of graphs which requires at least 4 colors for acyclic vertex coloring.
Hence, the assertion of Lemma 4.
Claim 1. C42Pk cannot be acyclically vertex colored with 3 colors for k ≥ 2.
To prove Claim 1, we use the simplest member of this family, C42P2 as shown in Figure 4.5.
Here note that, 1, 2, 3, 2 and 1, 2, 1, 3 are the only possible color patterns of 3 colors that can
be used to color a C4 such that it is properly and acyclically colored. Any other color combination
would just be a renaming of the colors. Also, the possible manners in which these colors can be validly
mapped to the second cycle such that they are also properly and acyclically colored are shown in Figure
4.5. One can do this by generating all the possbile permutations for the second cycle with these colors
38
2
3
1
3
1
2
3
2
1
2
3
2
3
1
2
1
1
2
3
1
3
1
2
3
2
3
1
2
3
1 2
1
(i) (ii)
(iv)(iii)
Figure 4.5 All possible distinct color combinations for C42P2 using 3 colors
and then see which ones agree to our above mentioned requirements.
By looking at Figure 4.5, one can see that there exists a bichromatic cycle of colors 2 and 3 in graph
(i). Similarly, there is a bichromatic cycle of colors 1 and 2 in both graphs (ii) and (iii), and, of colors
1 and 3 in graph (iv).
Hence, we conclude that a C42P2 cannot be acyclically colored with 3 colors. Or in other words, a
C42P2 requires at least 4 colors to be acyclically colored.
Using Claim 1, we have a family of graphs in the class Cm2Pk which requires at least 4 colors to
be acyclically colored. This coupled with Lemma 3 proves the assertion of Lemma 4 that a(Cm2Pk) =
4.
4.5.2 CYCLES 2 CYCLES
Now we consider the graphs when a member of CYCLES is taken product with another member
of CYCLES. These graphs look like their counterparts of the previous subsection i.e. CYCLES taken
product with PATHS, except for the fact that the multiple copies of the operand cycle would now be
39
connected cyclically, i.e. the corresponding vertices of the first and last copies would also be connected
in addition to the adjacent copies as before.
Here we note that this extra set of connections for Cm2Ck would break the coloring schemes for the
corresponding Cm2Pk. Most conspicuous among these is the instance where k is odd in which case the
coloring scheme would no longer remain proper let alone acyclic.
4.5.2.1 Cm2Ck
Lemma 5. All Cm2Ck can be colored with 7 colors.
Proof. We observe that the only edges in a Cm2Ck in addition to a Cm2Pk are the edges between the
corresponding vertices in the first and the last copy of Cm. This implies that any possible bichromatic
cycle in Cm2Ck when colored with the coloring scheme for Cm2Pk will use these edges. Hence, if
we color the last copy of Cm with colors that are different altogether from the ones used in other copies,
then such a scheme would acyclically color Cm2Ck.
Also, we know that any Cm can be acyclically colored with 3 colors. This gives us our first result on
the upper bound for number of colors to be used to color Cm2Ck acyclically.
a(Cm2Ck) ≤ a(Cm2Pk) + 3 (4.12)
Going by our scheme, we note that each vertex in the last copy has to be colored with a different scheme
to
1. maintain validity when k is odd.
2. maintain acyclicity when k is even.
This means that we must introduce atleast two more colors other than the ones previously used in
other copies. Also a third color is required to maintain acyclicity within this copty itself. This formu-
lation is true for both the cases whether m is even or odd. Figure 4.6 shows implementation of such a
scheme.
Hence, Cm2Ck (m,k > 2) can be colored with 7 colors.
40
3
2
1
2
4
3
3
1
1 4
6
5
5
6
7
Figure 4.6 A Cm2P3 colored with 7 colors, where m is odd
41
Chapter 5
Conclusions and Future Work
In this chapter, we discuss some remarks about the work presented thus far and possible directions
for future work.
5.1 Concluding Remarks
In this thesis, we have presented simple and effiecient linear time algorithms to acyclically edge color
three new classes of graphs which are K(p−1)(q−1),(p−1)(q−1), Kp(q − 1) and Kp(q−1)(r−1) for primes
p, q, r greater than 2. We have also provided acyclic vertex coloring schemes for some of the base cases
of the d−dimensional partial tori graphs, some of which are optimal. We here note the two important
aspects of the nature of our coloring schemes.
1. Even though some of the them are not very straightforward, they are all built upon one another in
a manner that exposes the structural properties of the graphs they are applied to.
2. All of our proofs are constructive in nature. Hence, they provide us with schemes that can be
applied on real graphs that can be used in several applications as mentioned in section 1.5.
5.2 Directions for Future Work
5.2.1 Edge Coloring
As mentioned before, our coloring schmes are very simple and flexible. Moreover, they build on one
another constructively. Hence, it maybe expected that these techniques can be applied to other classes
of graphs for which there are no efficient acyclic coloring schemes known yet.
42
5.2.2 Vertex Coloring
We expect that our work for the cases of graphs of the class d-dimensional partial tori can be ex-
tended to generalised results i.e. bounds on acyclic chromatic number of type G2K2, G2Pk (k > 2)
and G2Ck for any given graph G.
We also note that our coloring scheme for Cm2Ck seems to use the least number of colors possible.
It is therefore expected that our bound might turn out to be the optimal bound for these graphs. Also, it
is hoped that any rigorous proof of such said optimality could be extended to cover the other cases for
d-dimensional partial tori graphs.
43
Related Publications
1. Kishore Kothapalli, V.Ch. Venkaiah, Bharat Joshi, and K. Ramanjaneyulu, Acyclic Edge Color-
ing of Kp(q−1),Kp(q−1)(r−1), andK(p−1)(q−1),(p−1)(q−1) in International Conference on Graph
Theory and its Applications, Coimbatore, India, 2009.
2. Kishore Kothapalli, V.Ch. Venkaiah, Bharat Joshi, and K. Ramanjaneyulu, Acyclic Edge Coloring
of Kp(q−1) and Kp(q−1)(r−1), Journal of Discrete Algorithms (to appear).
3. Bharat Joshi, Kishore Kothapalli, On Acyclic Vertex Coloring of Grid like graphs, International
Conference on Recent Trends in Graph Theory and Combinatorics, Cochin, India, 2010 (to be
submitted).
44
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