1. Graph Theory Chapter 10 Coloring Graphs 大葉大學 (Da-Yeh Univ.) 資訊工程系 (Dept. CSIE) 黃鈴玲 (Lingling Huang)

2. Copyright  黃鈴玲 Ch10-2 Outline 10.1 Vertex Colorings 10.2 Chromatic Polynomials 10.3 Edge Colorings 10.4 The Four Color Problem

3. Copyright  黃鈴玲 Ch10-3 10.1 Vertex Colorings Focus: Partition the vertex (edge) set of an associated graph so that adjacent vertices (edges) belong to different sets of the partition.

4. Copyright  黃鈴玲 Ch10-4 Definition: A set S of vertices in a graph G is independent if no two vertices of S are adjacent in G. 應用 : (1) 化學藥品存放，避免交互作用。 (2) 水族館設計，會互相捕食的魚不能放同一個水族箱。 Definition: An independent set S of vertices in a graph G is called a maximal independent set if S is not a proper subset of any other independent set of vertices of G. The maximum cardinality of an independent set of vertices of G is called the independence number of G and is denoted by   (G).

5. Copyright  黃鈴玲 Ch10-5 Fig 10-1 A graph G with   (G) = 4,   (G) = 4,   (G) = 2. Definition: A clique in a graph G is a maximal complete subgraph. The maximum order of a clique is the clique number of G and is denoted by   (G).

6. Copyright  黃鈴玲 Ch10-6 Fig 10-1 A graph G with   (G) = 4,   (G) = 4,   (G) = 2. Definition: A set S of vertices in a graph G is a dominating set if every vertex not in S is adjacent to a vertex in S. A dominating set S is a minimal dominating set if no proper subset of S is also a dominating set. The domination number   (G) of G is the minimal cardinality of a dominating set of G.

7. Copyright  黃鈴玲 Ch10-7 Theorem 10.1: For every graph G,   (G)    (G). Proof. Let S be a maximum independent set of G.  |S| =   (G) Since S is maximum,  every vertex v  S must be adjacent to at least one vertex of S.  S is also a dominating set.    (G)  |S| =   (G)

8. Copyright  黃鈴玲 Ch10-8 The Independence Number Problem: For a given graph G of order p and positive integer k  p, is   (G)  k? Determining   (G),   (G), and   (G), are NP-complete problems. The Clique Number Problem: For a given graph G of order p and positive integer k  p, is   (G)  k? The Domination Number Problem: For a given graph G of order p and positive integer k  p, is   (G)  k?

9. Copyright  黃鈴玲 Ch10-9 不找最大的 independent set ，  改找較大的 independent set  改找最少組互斥的 independent set 使它們形成 V(G) 的 partition Definition: A coloring of a graph G is an assignment of colors (elements of some set) to the vertices of G, so that adjacent vertices are assigned distinct colors. If n colors are used, then the coloring is referred to as an n -cloloring of G. Every coloring of a graph G produces a partition of V(G) into independent sets, called color classes. If there exists an n -coloring of a graph G, then G is n - colorable.

10. Copyright  黃鈴玲 Ch10-10 Definition: The minimum n for which a graph G is n -colorable is called the chromatic number of G, and is denoted by  (G). A graph G with chromatic number n is also called n -chromatic. Fig 10-3 1 1 11 23 4  K   (G)  4  coloring  (G)  4  (G) = 4 G

11. Copyright  黃鈴玲 Ch10-11 The Chromatic Number Problem: For a given graph G of order p and positive integer b with 2 < b  p, is   (G)  b? It is also an NP-complete problem. (1)  (K p ) = p for every positive integer p. (2)  (C p ) = 3 if n  3 is odd.  (C p ) = 2 if n  4 is even. (3)  (K m,n ) = 2 for every pair m,n of positive integers.

12. Copyright  黃鈴玲 Ch10-12 Theorem 10.2: A graph is 2-colorable if and only if it is bipartite. Theorem 10.3: Let G be a connected graph with maximum degree  =  (G). Then (i)  (G)   (G)  1 + . (ii)  (G)   if and only if G is neither a complete graph nor an odd cycle. ( 此部分稱為 Brooks theorem)

13. Copyright  黃鈴玲 Ch10-13 Proof. (i)  (G)   (G), trivial. (i) 證明  (G)  1 +  There is a (1 +  )- coloring of G. Since each vertex has degree  . If there are 1+  colors, any vertex can be colored by a color different with all its neighbors.   (G)  1 +  (ii) 證明跳過

14. Copyright  黃鈴玲 Ch10-14 Example (1) K n  K 2 :  = n,  = n   = n (2) K n,n :  = 2,  = 2,  = n There are graphs for which  = . There are graphs for which  is arbitrarily large. There are also graphs for which  is large. Definition: A graph G is called F -free if G does not contain an induced subgraph isomorphic to F. Example: C n (n  4) and K m,n are triangle-free graphs.

15. Copyright  黃鈴玲 Ch10-15 Theorem 10.4: (Mycielski’s Theorem) For every positive integer n, there exists an n -chromatic, triangle-free graphs. Proof. (by induction on n ) (Basis) n =1: K 1 n =2: K 2 n =3: C 5 (Inductive) Assume that H is a triangle-free graph with  (H) = k, where k  3. We will show that there exists a triangle-free graph with chromatic number k +1.

16. Copyright  黃鈴玲 Ch10-16 Let G be a graph with V(G)= V(H) U { u 1, u 2, …, u p } U { u }. Suppose V(H) = {v 1, v 2, …, v p }. GH u u1u1 u5u5 v1v1 v5v5 E(G) = E(H) U { u i v j | v i v j  E(H), 1  i, j  p} U { u i u | 1  i  p} Fig 10-5 The Gr ö tzsch graph (  =4, triangle-free) It is clear that G is triangle-free. It remains to show that  (G)=k+1. Let c be a k -coloring of H. Let c’ be a coloring of G with c’(u i )=c(v i ),  i, c’(v i )=c(v i ),  i, and c’(u)=k+1.

17. Copyright  黃鈴玲 Ch10-17 It is clear that c’ is a ( k +1)-coloring of G. So  ( G )  k +1. If G has a k- coloring, say b, with colors 1, 2, …, k. Since u is adjacent to every vertex u i, b(u)  b(u i ),  i. Suppose b(u) = k, then the vertices u i are colored by 1, 2, …, k  1. Since  (H) = k, there are some vertices of H colored by k. Recolor each vertex v i having b(v i ) = k with the color b(u i ). A ( k  1)-coloring of H produced.    (G)=k+1

18. Copyright  黃鈴玲 Ch10-18 Algorithm 10.1 (Sequential Coloring Algorithm) [To produce a coloring of G with V(G) = { v 1, v 2, …, v p }.] 1. i  1 ( 正在 visit 的點是 v i ) 2. c  1 ( 預計著於點 v i 的顏色 ) 3.3.1 Sort the colors adjacent with v i in nondecreasing order and call the resulting list L i. 3.3.2 If c does not appear on L i, then assign color c to v i, and go to Step 5; otherwise, continue. 4. c  c +1 5. If i < p, then i  i +1, and return to Step 2; otherwise, stop.

19. Copyright  黃鈴玲 Ch10-19 Figure 10-6 G v1v1 v2v2 v5v5 v3v3 v6v6 v4v4 1 2 1 2 3 3 但實際只需 2 色 Alg 10.1 用了 3 色 點的編號會影響著色結果 For any graph G of order p, there are p ! possible ways to label the vertices. p ! 中必有一種編號方式可使 Alg 10.1 使用剛好  (G) 色

20. Copyright  黃鈴玲 Ch10-20 Figure 10-7 1 v1v1 v2v2 vnvn v3v3 u1u1 …… u2u2 u3u3 unun G: V(G) = {v 1, u 1, v 2, u 2, …, v n, u n }. E(G) = { u i v j | i  j } 1 22 33 n n 最糟時，可能給此種 bipartite graph 著 n 色

21. Copyright  黃鈴玲 Ch10-21 ※ Consider graphs whose chromatic number is decreased upon the removal of any vertex. Note that  (G  v) =  (G) or  (G)  1. Definition: A graph G is critically n -chromatic, n  2, if  (G) = n and  (G  v) = n  1 for every vertex v of G. Example: (1) C n (n is odd ) is critically 3-chromatic. (2) K n is critically n -chromatic. Note. Every graph with chromatic number n  2 contains a critically n -chromatic subgraph.

22. Copyright  黃鈴玲 Ch10-22 Theorem 10.5 Let G be a critically n -chromatic graph ( n  2). Then  (G)  n  1. Proof. Suppose v is a vertex of degree < n  1. G is a critically n -chromatic   (G  v) = n  1. There is an ( n  1)- coloring of G  v. deg( v ) < n  1  all neighbors of v are colored by at most n  2 colors. G has an ( n  1)- coloring. 

23. Copyright  黃鈴玲 Ch10-23 Homework Exercise 10.1: 1, 3, 7, 10, 14

24. Copyright  黃鈴玲 Ch10-24 Outline 10.1 Vertex Colorings 10.2 Chromatic Polynomials 10.3 Edge Colorings 10.4 The Four Color Problem

25. Copyright  黃鈴玲 Ch10-25 10.2 Chromatic Polynomials Note. Two colorings of a labeled graph G are considered different if they assign different colors to the same vertex of G. Focus. 計算有多少種不同的著色方法 Definition. The chromatic polynomial f ( G, t ) of graph G is the number of different colorings of G that use t or fewer colors. Note. If t <  (G), then f ( G, t ) = 0.

26. Copyright  黃鈴玲 Ch10-26 Figure 10-8 K4K4 uv x w t t  1 t  2 t  3 f (K 4, t ) = t (t   1)(t   2)(t   3) v1v1 v2v2 v4v4 v3v3 t t t t K4K4 f (K p, t ) = t (t   1)  (t   p+1) f (K 4, t ) = t 4 f (K p, t ) = t p

27. Copyright  黃鈴玲 Ch10-27 Figure 10-9 K 1,4 u1u1 u2u2 u4u4 u3u3 t  1 f (K 1,4, t ) = t (t   1) 4 t t  1 1  f (C 4, t ) = t (t   1) (t 2   3 t  + 3) u t t  1 v1v1 v2v2 v4v4 v3v3 C4C4 If c(v 2 ) = c(v 3 ): t  (t   1)  1  (t   1) t  2 If c(v 2 )  c(v 3 ): t  2 t  (t   1)  (t   2) 2

28. Copyright  黃鈴玲 Ch10-28 相當於 v1v1 v2v2 v4v4 v3v3 C4C4 If c(v 2 ) = c(v 3 ): If c(v 2 )  c(v 3 ): = v1v1 v 2 = v 3 v4v4 v1v1 v2v2 v4v4 v3v3

29. Copyright  黃鈴玲 Ch10-29 Definition. For a graph G with nonadjacent vertices u and v, Let G:u=v be a graph with V(G:u=v) = V(G)  {v}, E(G:u=v) = { e  E(G) | e is not incident with v} U { uw | vw  E(G) }. 即將 u 及 v 變成同一點 Theorem 10.6. If u and v are nonadjacent vertices in a, noncomplete graph G, then f ( G, t ) = f ( G + uv, t ) + f (G:u=v, t ) uv 著不同色 uv 著同色

30. Copyright  黃鈴玲 Ch10-30 Figure 10-11  f (P 4, t ) = t (t  1)(t  2)(t  3) + 3t (t  1)(t  2) + t (t  1) u v = u v + = u v + +2 = v u + 3+

31. Copyright  黃鈴玲 Ch10-31 Homework Exercise 10.2: 1