Edge Coloring of Split Graphs �

S. M. Almeida 1, C. P. de Mello 2

Institute of ComputingUNICAMP

Campinas, Brazil

A. Morgana 3

Department of MathematicsUniversity of Rome “La Sapienza”

Rome, Italy

Abstract

A split graph is a graph whose vertex set admits a partition into a stable set and aclique. The chromatic indexes for some subsets of split graphs, such as split graphswith odd maximum degree and split-indifference graphs, are known. However, forthe general class, the problem remains unsolved. This paper presents new resultsabout the classification problem for split graphs as a contribution in the directionof solving the entire problem for this class.

Keywords: edge coloring, split graphs, classification problem.

� This research was partially supported by CAPES and CNPq (300934/2006-8 and470420/2004-9).1 Email: sheila@ic.unicamp.br2 Email: celia@ic.unicamp.br3 Email: morgana@mat.uniroma1.it

Electronic Notes in Discrete Mathematics 30 (2008) 21–26

1571-0653/$ – see front matter © 2008 Elsevier B.V. All rights reserved.

www.elsevier.com/locate/endm

doi:10.1016/j.endm.2008.01.005

1 Introduction

A k-edge coloring of a graph G is an assignment of one of k colors to eachedge of G such that there are no two edges with the same color incident toa common vertex. In the discussion below, a “coloring” of a graph alwaysmeans an edge coloring, while a “k-coloring” is a coloring that uses only kcolors. A k-coloring partitions the set of edges of G into k color classes. Thechromatic index , χ′(G), of G is the minimum k such that G has a k-coloring.By definition, χ′(G) ≥ Δ(G), where Δ(G) is the maximum degree of G.

In 1964, Vizing [7] showed that for any simple graph G, χ′(G) ≤ Δ(G) +1. It was the origin of the Classification Problem, that consists of decidingwhether a given graph G has χ′(G) = Δ(G) or χ′(G) = Δ(G) + 1. In the firstcase, we say that G is Class 1, otherwise, we say that G is Class 2. Despitethe powerful restriction imposed by Vizing, it is very hard to compute thechromatic index in general. In fact, it is NP-complete to decide if a graph isClass 1 whereas Class 2 recognition is co-NP-complete [4]. In 1991, Cai andEllis [1] proved that this holds also when the problem is restricted for someclasses of graphs.

A split graph is a graph whose vertex set admits a partition into a stable setand a clique. The chromatic indexes for some subsets of split graphs, such assplit graphs with odd maximum degree [2] and split-indifference graphs [5], areknown. However, for the general class, the problem remains unsolved. Thispaper presents new results about the classification problem for split graphs asa contribution in the direction of solving the entire problem for this class.

2 Definitions and Necessary Background

In this paper, G denotes a simple, finite, undirected and connected graph;V (G) and E(G) are the vertex and edge sets of G. Write n = |V (G)| andm = |E(G)|. The maximum degree of G is Δ(G) and, when there is noambiguity, we use the simplified notation Δ. A Δ(G)-vertex is a vertex of agraph G with degree Δ(G). A universal vertex is a vertex with degree n − 1.Let v be a vertex of G. The set of vertices adjacent to v in G is denoted byN(v), and N [v] = {v}∪N(v). A clique is a set of pairwise adjacent vertices of agraph. A maximal clique is a clique that is not properly contained in any otherclique. A stable set is a set of pairwise no adjacent vertices. A subgraph of Gis a graph H with V (H) ⊆ V (G) and E(H) ⊆ E(G). For X ⊆ V (G), denoteby G[X] the subgraph induced by X, that is, V (G[X]) = X and E(G[X])consists of those edges of E(G) having both ends in X. Let D ⊆ E(G). The

S.M. Almeida et al. / Electronic Notes in Discrete Mathematics 30 (2008) 21–2622

subgraph induced by D is the subgraph H with E(H) = D and V (H) is theset of vertices v having at least one edge of D incident to v. We denoted byKn a complete graph with n vertices.

The following lemmas are used in our discussion about the coloring of splitgraphs.

Lemma 2.1 [3] The complete graph Kn is Class 1 if, and only if, n is even.

Lemma 2.2 [3] Every bipartite graph is Class 1.

A graph G is overfull when m > Δ(G)⌊

n2

⌋and n is odd. In [6], Planthold

shows that a graph G with n odd and containing a universal vertex is Class 1if, and only if, G is not overfull.

An equitable k-coloring of a graph G is a k-coloring of G such that thesizes of any two color classes differ by at most one. We say that a vertex vmisses a color c (or that a color c misses a vertex v) when there is no edgewith color c incident to v. Otherwise, we say that the color c appears in v.

Lemma 2.3 Let n be an odd integer. If Kn is colored with n colors, then eachone of these n colors misses exactly one vertex and each vertex misses exactlyone color.

Lemma 2.4 Let n be an even integer. Then Kn has an equitable n-coloringsuch that each vertex misses one color, each one of n

2colors misses two ver-

tices, and the other n2

colors appears in every vertex of Kn.

Lemma 2.5 Let n be an even integer and G = Kn \ F , where F is a subsetof E(Kn) with |F | = k. Then G has an equitable (n − 1)-coloring, such thatthere are k′ = min{k, n − 1} colors missing at least two vertices of G.

Lemma 2.6 Let n be an odd integer and G = Kn \ F , where F is a subsetof E(Kn) with |F | = k, k ≥ n−1

2. Then G has an equitable (n − 1)-coloring,

each one of the min{k − n−12

, n−12} colors misses at least three vertices of G,

and each one of the remaining max{0, 3(n−1)2

− k} colors misses at least onevertex of G.

3 Coloring Split Graphs

A split graph is a graph that admits a partition {Q, S} of its set of vertices suchthat Q is a clique and S is a stable set. The classification problem is solved forsome subclasses of split graphs [5,2]. In this work, we solve the classificationproblem for a new subset of split graphs, presented in Theorem 3.1. This

S.M. Almeida et al. / Electronic Notes in Discrete Mathematics 30 (2008) 21–26 23

result is strongly based on the work of Planthold [6]. In the following, weassume that Q is a maximal clique.

Theorem 3.1 Let G be a split graph with even maximum degree. If existsa Δ(G)-vertex v such that N [v] admits a partition {L, R} where |R| = Δ(G)

2,

R ⊂ Q, the vertices in L are not adjacent to vertices in V (G)\N [v], G[L] hask edges, k ≥ Δ

4, and R has at most k′ Δ(G)-vertices, k′ = min{k, Δ

2}, then G

is Class 1.

Proof. Let G be a split graph with partition {Q, S} and Δ = Δ(G) even.Suppose that there exist a Δ-vertex v of G as described above. Let P =V (G)\{L∪R}. (See Fig. 1.) Since, by hypothesis, |R| = Δ

2, then |L| = Δ

2+1.

Hence, the maximum degree of G[L] is at most Δ2. We consider two cases: Δ

2

is odd and Δ2

is even.

L

R

P

V

Fig. 1. A split graph G and the subsets L, R and P of V (G).

Case 1: Δ2

is odd

The graph G[L] is isomorphic to a subgraph of KΔ2

+1 and, by hypothesis,

G[L] has k ≥ Δ4

edges. By Lemma 2.5, G[L] has an equitable Δ2-coloring where

each color ci misses at least two vertices in L, 1 ≤ i ≤ k′ = min{k, Δ2}.

Let R = {v1, v2, . . . , vΔ2}, and let J = {v1, v2, . . . v|J |} be the subset of

vertices of R that are adjacent to every vertex of L. The vertices in J areadjacent to Δ

2− 1 vertices in R and Δ

2+ 1 vertices in L, therefore these

vertices have degree Δ. By hypothesis, there are at most k′ Δ-vertices inR, so |J | ≤ k′. The graph G[R] is isomorphic to KΔ

2and Δ

2is odd, hence,

by lemmas 2.1 and 2.3, R can be colored with Δ2

colors such that each colormisses exactly one vertex and each vertex misses one color. By the symmetryof G[R], we can perform the coloring of G[R] such that the color missed byvertex vi is ci, 1 ≤ i ≤ Δ

2. Since |J | ≤ k′ and the vertices in J are adjacent

to every vertex in L, each vertex vi in J is adjacent to a vertex u in L thatmisses the color ci. Assign the color ci to the edge {vi, u}. For each vertex v

S.M. Almeida et al. / Electronic Notes in Discrete Mathematics 30 (2008) 21–2624

in R\J with degree Δ2

+ 1, there is a vertex w in P such that w is adjacentto v. So, assign the color c, missed by v in the coloring of G[R], to the edge{v, w}. This process can be repeated for every vertex in R\J that is adjacentto Δ

2+ 1 vertices in L ∪ P because the color missed by each vertex in R is

distinct of the other ones.

By hypothesis, the vertices in L are not adjacent to vertices in V (G)\V (H).Hence the graph induced by the uncolored edges of G is a bipartite graph andits maximum degree is at most Δ

2. Therefore, by Lemma 2.2, we can color this

subgraph with Δ2

new colors.

Case 2: Δ2

is even

In this case, G[L] is isomorphic to a subgraph of KΔ2

+1 with Δ2

even,

|E(G[L])| = k and k ≥ Δ4. So, by Lemma 2.6, G[L] has an equitable Δ

2-

coloring such that each one of p = min{k − Δ4, Δ

2} colors misses at least 3

vertices in L and each one of the other Δ2− p colors misses at least 1 vertex

in L. Let c1, c2, . . . , cp be the colors missed by at least 3 vertices in L.

The graph G[R] is isomorphic to KΔ2

and Δ2

is even. So, by Lemma 2.4,

G[R] has an equitable Δ2-coloring such that each one of Δ

4colors misses two

vertices, each one of the other Δ4

colors does not miss any vertex in R, andeach vertex in R misses exactly one color. The set R is divided in threesubsets namely J , J ′, and N , where: J = {v1, . . . , v|J |} is the set of the Δ-vertices in R that are adjacent to every vertex in L; J ′ = {v|J |+1, . . . , v|J |+|J ′|}is the set of the Δ-vertices in R that are adjacent to at least one vertex inP = V (G)\{L ∪R}; and N = {v|J |+|J ′|+1, . . . , vΔ

2} is the set of the vertices in

R that are not Δ-vertices. Note that each one of this sets can be an emptyset, and |J | + |J ′| + |N | = Δ

2.

The symmetry of R allows us to choose which vertex in R misses a specificcolor. Let p′ = min{p, Δ

4}, and let X = {v1, . . . , v2p′} be the set of vertices in

R such that vertices v2i−1 and v2i miss the color ci, 1 ≤ i ≤ p′. Note that Xcan be empty. By hypothesis, there are at most k′ = min{k, Δ

2} Δ-vertices in

R, so at least Δ2−k′ vertices in R are not Δ-vertices. Let Z = {vk′+1, . . . , vΔ

2}

be the set of these vertices and Y = {v2p′+1, . . . , vk′} = R \ (X ∪Z). The setsX, Y , and Z are pairwise disjoint and |X| + |Y | + |Z| = Δ

2.

If k ≥ Δ2, then X = R and the sets Y and Z are empty. If Δ

4≤ k < Δ

2,

then Z and Y have size Δ2−k′ > 0. In this case, let α = {cp′+1, . . . , cΔ

4}. Each

color of α has to miss two vertices in R and each vertex in Y ∪ Z has to missone color. Since |α| = |Y | = |Z| = Δ

2−k′, for each color c in α, we choose one

vertex in Y and one vertex in Z to miss the color c. Now, no two vertices ofY miss the same color.

S.M. Almeida et al. / Electronic Notes in Discrete Mathematics 30 (2008) 21–26 25

If X is nonempty, for each vertex v in X that misses a color c and hasΔ2

+ 1 uncolored edges incident to it, we color one of these edges as follows. Ifv belongs to J , we choose a vertex u that belongs to L and misses the color c,and we assign the color c to the edge {v, u}. If v belongs to J ′, there are twocases. If v is adjacent to a vertex w in P such that the color c is not incidentto w, then we assign the color c to the edge {v, w}. Otherwise, v is adjacentto a vertex u that belongs to L and misses the color c, so we assign the colorc to the edge {v, u}. If Y is nonempty, we can color one edge incident to eachΔ-vertex that is in Y . For each Δ-vertex vi in Y , choose a neighbor w in P ,and assign the color missed by vi to {vi, w}, 2p′ + 1 ≤ i ≤ k′. Remind thatthere are no Δ-vertices in Z, so each vertex of Z has at most Δ

2uncolored

incident edges. Now, each Δ-vertex in R has at most Δ2

uncolored incidentedges. The graph induced by the uncolored edges of G is a bipartite graphwith a partition {L ∪ P, R} and maximum degree Δ

2. So, by Lemma 2.2, we

can color this subgraph with Δ2

new colors.

Therefore, by cases 1 and 2, we conclude that G is Class 1. �

We believe that the result of Theorem 3.1 can be extended to other subsetsof split graphs. It is easy to see that a split graph with odd Δ

2containing more

than k′ Δ-vertices in R but with at most k′ vertices adjacent to every vertexin L, while the other conditions of Theorem 3.1 are satisfied, is Class 1.

References

[1] Cai, L., and J. A. Ellis, NP-completeness of edge-colouring some restricted graphs.Discrete Applied Mathematics, 30:15–27, 1991.

[2] Chen, B-L., H-L. Fu, and M. T. Ko, Total chromatic number and chromatic index ofsplit graphs. Journal of Combinatorial Mathematics and Combinatorial Computing,17:137–146, 1995.

[3] Fiorini, S., and R. J. Wilson, Edge-colourings of graphs. Research Notes inMathematics, Pitman, 1977.

[4] Holyer, I., The NP-completeness of edge-coloring. SIAM Journal on Computing,10:718–720, 1981.

[5] Ortiz, C., N. Maculan, and J. L. Szwarcfiter, Characterizing and edge-colouring split-indifference graphs. Discrete Applied Mathematics, 82:209–217, 1998.

[6] Plantholt, M., The chromatic index of graphs with a spanning star. Journal of GraphTheory, 5:45–53, 1981.

[7] Vizing, V. G., On an estimate of the chromatic class of a p-graph. Diskret. Analiz.,3:25–30, 1964.

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