Graphs & Networks (MATH F243)Dr. Krishnendra Shekhawat

• Handout

• Quizzes (10 Marks, 10 Minutes, Best 4 out of 5)

• Tutorial Sheets (Nalanda)

• Textbook

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Textbook

G. Agnarsson and R. Greenlaw, Graph Theory: Modeling, Applications, and Algorithms,

Pearson, 2007.

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Chapter 1 Introduction to Graph Theory

Introduction

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The theory of graphs is one of the few fields of mathematics with a definite birth date (Oystein Ore, author of first graph theory book written in English).

Graph theory is considered to have begun in 1736 with the publication of Euler’s solution of Kőnigsberg Bridge Problem (Leonard Euler, known to be the Father of Graph Theory).

Two hundred years later, in 1936, Denes Konig wrote the first book on graph theory.

Graph Theory branches – Theoretical, Algorithmic, Algebraic

1.2 Why Study Graphs?• Problem 1.1: The Bridges of Kőnigsberg (1736 Leonard Euler)

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Problem: Make a round trip through downtown Kőnigsberg, traversing each bridge exactly once.

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Ans: Euler proved that this problem has no solution. (Chapter 5)

Try to have even degree for all vertices and get a solution.

Q: Start at a vertex, go along each edge exactly once, and end up at the starting vertex？

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Problem 1.3: Job Assignments

Jobs:

Applicants:

qualified

Bipartite Graph

Problem 1.8: Is the company able to meet its hiring need? If so, provide a possible set of hires that meet their needs.

Ans: No (Ch10 Matching)

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Problem 1.4: Storing Volatile Chemicals

C1, C2, …, C7 are different kinds of chemicals where an edge between Ci and Cj indicates a grave danger in storing these chemicals in the same warehouse.

Q. What is the minimum number of warehouses the factory needs in order to store its chemical products safely？

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Problem 1.4: Storing Volatile Chemicals

Problem: What is the minimum number of warehouses the factory needs in order to store its chemical products safely？Ans: 4 (Ch8: Graph Coloring)

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1.4 The Definition of a Graph• A graph or a general graph is an ordered triple G = (V, E, ), where i. V (empty set). ii. VE = . iii. (edgemap): E P(V) is a map such that |(e)| {1, 2} for each

eE.

• Elements of V are called vertices of G (Vertex set V(G))• Elements of E are called edges of G (Edge set E(G))• Vertex and edge are denoted by v and e respectively• Vertices in (e) are called endvertices of the edge e• If both V and E are finite then G is finite graph, if V or E is infinite then G is called infinite (unless otherwise stated, a graph always mean finite graph).

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Example

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G=(V, E, )V={u1, u2, u3, u4, u5}E={e1, e2, e3, e4, e5, e6}

(e1)={u1, u2}(e2)=(e3)={u1, u3}(e4)={u2, u3}(e5)={u3, u4}(e6)={u4}

u5 is called isolated.

Compute (ei) for all i

Ch1-13

Definition of a graph

• A graph G is a finite nonempty set V(G) of vertices (also called nodes) and a (possibly empty) set E(G) of 2-element subsets of V(G) called edges (or lines).

• V(G) : vertex set of G• E(G) : edge set of G• Edge {u, v} = {v, u} = uv (or vu)

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Order, Size, Adjacency• The number of vertices in a graph G is called its order (denoted by

|V(G)| ).• The number of edges is its size (denoted by |E(G)|).• Two vertices u and v are adjacent (or neighbors) if they are endpoints of some edge eE.• Two edges e and f are adjacent if they have a common endvertex. • u and e are incident if u is an endvertex of e. • Loop: A loop is an edge whose endvertices are equal.• E’ is a set of multiple or parallel edges if all |E’|≥2 and e’E’ have the same set of endvertices.• u is called isolated if it is not an endvertex of any edge.

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Example

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Endvertices of e2: u1, u3u1, u2: adjacentu1, u4: not adjacente1, e2: adjacente1, e5: not adjacente3, e2: parallelu4 is adjacent to itselfe6 is adjacent to itself

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Q. Show that any graph with at least 6 vertices contains 3 vertices that are pairwise adjacent, or 3 vertices that are pairwise non-adjacent.

Pick an arbitrary vertex v1. There are at least 5 other vertices, such that either v1 has at least 3 neighbors, or it has at least 3 non-neighbors. In the first case, label the neighbors v2, v3, v4. If any two of v2, v3, v4 are adjacent, then with v1 they form a pairwise adjacent triple; otherwise, no two are adjacent, so v2, v3, v4 form a pairwise non-adjacent triple. Similarly, in the case where v1 has 3 non-neighbors v2, v3, v4, we get a pairwise non-adjacent triple if any 2 of v2, v3, v4 are not adjacent, and otherwise v2, v3, v4 are pairwise adjacent.

1.5 Examples of Common Graphs• Simple graph: A graph having no multiple edges or loops.

• Null graph

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Path graph

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Cycle graph

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Complete graph

Complete graph K7

A graph G is bipartite if V(G)=V1 V2 such that every edge of G

joins a vertex of V1 and a vertex of V2.

G: v1 v6

v4 v5 v3 v2

redrawn of G

V1

V2

disjoint union

v1v2

v4

v3

v6v5

+

Q. Is the graph G given below bipartite.

Bipartite graph

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Complete bipartite graph

Questions

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Q1. Determine whether the graph below decomposes into copies of P4.

Q2. (Puzzle) For some n, give a graph with n vertices, n + 3 edges, and exactly 6 cycles.

Result. A graph is bipartite if and only if it has no odd cycles (will be proved in Chapter 4).

Hand-Shaking

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Consider a collection of guests at a party. Suppose some guests shook hands with some other guests. If we asked everyone at the party how many guests they shook hands with and added those numbers all up, this sum would be equal to

twice the number of total hand shakes.

Is it possible that every two of them are acquainted with a different number of people in the group?(Suppose if A knows B, then B knows A.)

1.6 Degrees and Regular Graphs

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f u or hborsis the

N(u4) = N[u4] = {u3, u4}

N(u1) = {u2, u3}

N[u1] = {u1, u2, u3}

Q. Compute N(u1), N[u1] and N(u4)

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degree sum = 12E(G) = 6

The first theorem of graph theory

Proof. Each edge contribute twice to the degree sum.

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RemarksThe null graph Nn is 0-regular.The cycle Cn is 2-regular.The complete graph Kn is (n-1)-regular.The complete (m,n)-bipartite graph Km,n is a regular graph if and only if m=n (in this case, it is m-regular).

Regular graph

Q. Compute the number of edges in k-regular graph on n vertices and the complete graph Kn . (Corollary 1.30)Every k-regular graph on n vertices has kn/2 edges. In particular, the complete graph Kn has (n – 1)n/2 edges.

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Q. Describe all 1-regular simple graphs and all 2-regular simple graphs. Show that a simple graph on an odd number of vertices cannot be 3-regular, while for every even n ≥ 4, there is a 3-regular simple graph on n vertices.

A 1-regular graph is just a disjoint union of edges.

A 2-regular graph is a disjoint union of cycles.

Suppose there is a 3-regular graph G on 2k + 1 vertices. Since 2|E(G)|=sum of the degrees=3(2k+1)=6k+3, which is a contradiction.For n = 4, K4 is the only 3-regular graph with 4 vertices. We now construct a 3-regular graph on 2m vertices, for each m ≥ 3. We take two cycles u1u2 … um, and v1v2 … vm of the same length, and we place an edge between ui and vi for each 1 ≤ i ≤ m. The resulting graph is indeed 3-regular, since ui is adjacent to ui−1, ui+1, and vi , and similarly for vi .

Q.

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Q. Every vertex of a graph G of order 14 and size 25 has degree 3 or 5. How many vertices of degree 3 does G have?

Q. A graph G of order 7 and size 10 has six vertices of degree a and one of degree b. What is b? Draw the graph G.

sol. Suppose there are x vertices of degree 3, then there are 14-x vertices of degree 5. Now |E(G)| = 25 degree sum = 50 3x + 5(14-x) = 50 x = 10

sol. 6a + b = 20 (a, b) = (0, 20) () (1, 14) () (2, 8) () (3, 2) () a=3, b=2.

Degree SequenceLet G=(V, E), V={v1, v2, …, vp}.

Then s: deg(v1), deg(v2), …, deg(vp) is called a degree seqence of G. For convenient, assume s is non-increasing, then s is unique.

s : 3, 3, 2, 1, 1, 0G 3 2

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minimum degree d(G)

maximum degree D(G)

Graphical SequencesWe call a sequence of nonnegative integers graphical if it is the degree

sequence of some graph.

Theorem (Havel-Hakimi)Let s be a non-increasing sequence: d1, d2, …, dp, where di N, i, p ≥ 2, d1

≥ 1.Let s1 be the sequence:

Then s is graphical if and only if s1 is graphical.

d2-1, d3-1, …, dd1+1-1, dd1+2, …, dp

Q. Check if degree sequence s: 4, 4, 3, 3, 2, 2 is graphic.

s: 4, 4, 3, 3, 2, 2s1

’: 3, 2, 2, 1, 2 (delete first term (4) and subtract 1 from next 4)s1: 3, 2, 2, 2, 1 (reorder)s2: 1, 1, 1, 1 (delete first term and subtract 1 from next 3)s3

’: 0, 1, 1 (delete first term)s3: 1, 1, 0 (reorder)s4: 0, 0 (delete first term ) s is graphical

Algorithms: d1, d2, …, dp be the sequence of integers. To determine whether s is

graphical:

• If di = 0, i, then s is graphical. • If di<0 for some i, then s is not graphical.Otherwise, go to Step 3.• Reorder s to a non-increasing sequence, if necessary.• Delete the first number, say n, from the sequence and subtract 1 from

the next n numbers in the sequence. Return to Step 1.

Q. Draw the graph for degree sequence s: 4, 4, 3, 3, 2, 2.

s4 : 0, 0 s3 : 1, 1, 0 s2 : 1, 1, 1, 1 s1 : 3, 2, 2, 2, 1 s: 4, 4, 3, 3, 2, 2

G

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2 3

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s: 5, 4, 3, 2, 1, 1s1: 3, 2, 1, 0, 0 (delete 5)s2: 1, 0, -1, 0 (delete 3) s is not graphical

Q. Check if degree sequence s: 5, 4, 3, 2, 1, 1 is graphic.

1.7 Subgraphs

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is a subgraph of G if

We write G’ G.

Q. Check if G’ G.

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Induced Subgraph (Vertex-Induced Subgraph)

Q. Derive G[u1, u2, u3].

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There are 4 vertices, so there are 24 – 1 induced subgraphs.We do not consider as a subgraph.

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Edge-Induced Subgraph

Q. Derive G[e1, e2, e3, e4].

Q. Let G be a labeled (p, q) graph. How many different edge-induced subgraphs does G have?

2q-1 ( X E(G) but X )

1.8 The Definition of a Directed Graph

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u

ve

e'

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G = (V, E, h)

V = {u1, u2, u3, u4, u5}

E = {e1, e2, e3, e4, e5, e6}

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If G = (V, E, h) is a directed graph, then in a natural way we can form the corresponding underlying graph G = (V, E, ), where h (e) = (u, v) implies (e) = {u, v}.

Conversely, forming a directed graph from an undirected graph can not be done uniquely. In fact, if G has exactly m nonloop edges, then there are 2m choices for forming a directed graph G from G.

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Path digraph

Cycle digraph

Simple digraph: a digraph without directed loops and parallel directed edges.

1.9 Indegrees and Outdegrees in a Digraph

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N +(u3) ={u1, u4}

N -(u3) ={u1, u2}

and the

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Q. Let G be a digraph in which indegree equals outdegree at each vertex. Prove that there exist a cycle in G.

A finite directed graph contains a (directed) cycle if every vertex is the tail of at least one edge (has positive outdegree). (The same conclusion holds if every vertex is the head of at least one edge.)

Let G be such a graph, let P be a maximal (directed) path in G, and let x be the final vertex of P.

Since x has at least one edge going out, there is an edge x y. Since P cannot be extended, y must belong to P. Now x y completes a cycle with the y, x­subpath of P.