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VERTEX COLORINGSMateri Diskusi Kuliah

Teori Graf

PRODI PENDIDIKAN MATEMATIKAFKIP UNIVERSITAS JEMBER

KELOMPOK 1

OLEH :

Introduction of Vertex coloring

Definition of vertex

coloring

Example of Vertex

Coloring

Outline

Chromatic Number

Critical Graph

Theorem and Proof

Outline

Vertex Coloring

Given a graph G, find a

coloring of the vertices so that

no two neighbors in G have the same

color

Improper Coloring

Proper Coloring

INTRODUCTION

The wheel with n spokes, is the graph that consist of an n – cycle and one additional vertex that is adjacent to all the vertices of the cycle.. In Figure 2.1.1 we display and

Vertex Coloring

nw

4W 5W

INTRODUCTION

1

12

2

3

4W1

1

2

23

4

5WFigure 2.1.1

DEFINITION OF VERTEX COLORING

Vertex Coloring

Given a graph G, we define a coloring of G to be an

assignment of colors to the vertices of G such that no two adjacent vertices receive the

same color.

EXAMPLE OF VERTEX COLORING

EXAMPLE OF VERTEX COLORING

If G has p vertices we can color G with p colors.

We denote the minimum number of colors necessary to color G by X( G ), the chromatic number of the graph G.

X( G ) = n means the graph G is colorable by n colors, and G is not colorable by n – 1 colors. Example in figure 2.1.1, X ( ) = 3 and X ( ) = 4.

What is

chromatic number

???

CHROMATIC NUMBER

4w5w

The only graph with p vertices and chromatic number equal to p is

For Instance, what is the chromatic number of the graph G in figure 2.1.2 ?

We see that there is a subgraph isomorphic to K4. Thus

What is

chromatic number

???

CHROMATIC NUMBER

PK

4 )( GX

b

a c

e

Figure 2.1.2

CHROMATIC NUMBER

11

22

3

4

3

5

CHROMATIC NUMBER

Figure 2.1.4Figure 2.1.3

Let G be a graph, and let v be a vertex of G. If we remove v and all the edges incident with v from G, the resulting graph is called G – v. Similarly, if e is any edge of G, the graph obtained from G by removing e is G – e. We leave all the vertices in G when we remove an edge. If v is a vertex of , then is isomorphic to . If G is the graph in figure 2.1.2, G – a is the graph in Figure 2.1.3 and G – e is the graph in Figure 2.1.4

PKvKP 1PK

CHROMATIC

NUMBER

Critical GraphLet G be a graph. If H is a subgraph of G, and H ≠ G, then H is called a proper subgraph of G. if χ(H) < χ(G) for every proper subgraph H of G, then we say that G is critical.The graphs of Figures 2.1.2 and 2.1.5 are critical, and the complete graph Kn is also critical for every n.

DEFINITIONFigure

2.1.5

Theorem 2.1.1

THEOREM

Every critical graph is connected.

Theorem

2.1.2

THEOREM

Every graph G contains a critical subgraph H such that χ(H) = χ(G).

Prove Theorem

2.1.2

THEOREM

1. If G is critical, then the theorem is true for H = G.

2. If G is not critical, then there exists a proper subgraph of H1 of G with χ(H1) = χ(G).

3. If H1 is critical, then we are done; H = H1.

4. If H1 is not critical there exists a proper subgraph of H2 of H1 such that χ(H2) = χ(H1) = χ(G).

5. We continue in this manner. Since G is finite, for some k we must obtain a subgraph Hk such that Hk is critical. Then H = Hk.

Theorem 2.1.3.a

THEOREM

If G is critical with chromatic number four, then the degree of each vertex is at least three.

Prove Theorem 2.1.3.a

THEOREM

1. Suppose that the theorem is false.2. Then there exists a critical graph G with

chromatic number four and a vertex v of G such that the degree of v is at most two.

3. Since G is critical, G − v can be colored with only three colors.

4. So we color G − v with three colors.5. We put the vertex v back; v is adjacent

to at most two vertices, so there is at least one of the three colors left to color v.

6. But this is a contradiction, since G has chromatic number four.

7. Thus the degree of each vertex is at least three.

Theorem 2.1.3.b

THEOREM

If G is critical with chromatic number χ, then the degree of each vertex is at least χ – 1.

Prove Theorem 2.1.3.b

THEOREM

1. The proof is the same with theorem 2.1.3.a. ; instead of four we substitute χ.

2. Another example that illustrates the more general theorem is the critical graph of Figure 2.1.2, where χ = 5, and the degree of each vertex is at least four.

Theorem

2.1.4.

THEOREM

If G is a critical graph with p vertices and q edges, and G has chromatic number χ, then the relation

(χ − 1)p ≤ 2qholds.

Prove Theorem

2.1.4

THEOREM

1. By theorem 2.1.3, the degree of each vertex of G is at least χ – 1, and there are p vertices.

2. So the sum of the degrees of the vertices of the G is at least (χ – 1)p.

3. By theorem 1.1.1, the sum of the degrees of the vertices of G is equal to 2q.

4. Hence theorem 2.1.4 is proved.

The graph in Figure 2.1.7 is also critical. There exists graphs of arbitrarily high girth and arbitrarily high chromatic number [7],[17].

The girth of a graph is the length of a shortest cycle in

the graph. The girth of the graphs in Figure 2.1..6 and

2.1.7 is four.

We know that if a graph G contains a subgraph isomorphic to K4 or Wn for n odd, then χ(G) ≥ 4.

The graphs in Figures 2.1.6 and 2.1.7 also have chromatic number 4, but they contain no triangles, and that makes it hard to check that the two graphs are not colorable by three colors.

0101

0303

02020404 NOTE :

Figure 2.1.6 The Grotzsch Figure 2.1.6 The Grotzsch

Figure 2.1.7

Theorem

2.1.5.

THEOREM

(ErdÖs-Lovász). For every two integers m, n ≥ 2, there exists a graph with chromatic number n whose girth exceeds m.

Let G be a graph, and let x and y be vertices of G. The distance from x to y in G, denoted by d(x,y), is the length of a shortest path from x to y. If there is no path from x to y, we say d(x,y) = ∞. In Figure 2.1.2, d(a,b) = 2, d(a,c) = 1.

The cube graph Q3 of Figure 1.2.5 is another example of a bipartite graph, but the dodecahedron graph is not. 

An unsolved coloring problem is the following conjecture of Lovász. If G is graph such that χ(G – v – w) = χ(G) – 2 for every pair of adjacent vertices v and w, then G is a comlete graph. Exersize 2.1.3 is a modified version of this problem. It differs by only one word, but it is much easier to prove.

A graph G is called bipartite if χ(G) ≤ 2. A special example of a bipartite graph, the complete bipartite graph Km,n, was defined in chapter 1.

0101

0303

02020404 NOTE :

Theorem

2.1.6.

THEOREM

graph G is bipartite if and only if every cycle in G has even length.

Prove Theorem

2.1.6

THEOREM

1. Assume that G is bipartite. Suppose that G contains an odd cycle C. Then χ(C) = 3. Thus χ(G) ≥ 3, but this is a contradiction since G is bipartite. Hence G can not contain an odd cycle.

2. Now assume that G is a graph with no odd cycles. Without loss of generality, we assumed that G is connected. Why can we do this?

3. Select a vertex x0 of G. We color G as follows. If x is a vertex of G, we color x red if d(x0,x) is even, and we color x blue if d(x0,x) is odd. (So for instance, since d(x0,x0) = 0, x0 is colored red because 0 is even.)

4. Since every distance is either even or odd, every vertex is now colored.

Prove Theorem

2.1.6

THEOREM

5. Now we must show that no two adjacent vertices have the same color. We consider two adjacent vertices x and y.

6. We choose a shortest path from x0 to x and a shortest path from x0 to y. let u be the last common vertex in these shortest paths. (See Figure 2.1.8.) The vertex u may be equal to x0, or u may also be x or y.

Prove Theorem

2.1.6

THEOREM

7. Now we consider d(u,x) and d(u,y). If u is one of x or y, then either d(u,x) = d(u,y) + 1 or d(u,x) = d(u,y) – 1. In either case, one of the distances is odd and one is even. We say they have different parity.

8. If u is not one of x or y, we now compute the length of the cycle in Figure 2.1.8. It is d(u,x) + 1 + d(u,y). We know this is even. Hence d(u,x) and d(u,y) have different parity.

9. Since d(x0,x) = d(x0,u) + d(u,x) and d(x0,y) = d(x0,u) + d(u,y), d(x0,x) and d(x0,y) also have different parity. Thus x and y receive different colors, and χ(G) ≤ 2, so G is bipartite.

THEOREM

NOTE

Theorem 2.1.6 shows that trees are bipartite. Every cycle in a tree has even length, since there are no cycles in a tree. In other words, if there is a cycle in a tree, then the cycle has even length. An “if-then” statement is always true if there is no circumstance that makes the “if” part true. Such a statement is vacuously true. We give you a strange example. All my violins have length. This is true since I have no violins. See also exercise 2.1.8.

THEOREM

NOTE

The diameter of a graph G is the maximum distance between any two vertices of G. For instance, the diameter of Pn is n; the diameter of Wn is 2 if n > 3; the diameter of Kn is 1; and the diameter of Km,n is 2 if m or n is at least two. In applications of graph theory to communications networks, diameter is important. One wishes to have a graph with small diameter and as few edges as possible, together with certain other restrictions. This is a very active area of research now.

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