Graph Theory Chapter 8 Graph Coloring 2010 12

Graph Theory Chapter 8 Graph Coloring 大葉大學 資訊 程系 黃鈴玲 2010. 12

Contents Ø Ø Ø 8. 1 8. 2 8. 3 8. 4 8. 5 The Chromatic Number of a Graph Multipartite Graphs Results for General Graphs Planar Graphs Edge Coloring of a Graph 2

8. 1 The Chromatic Number of a Graph Definition 8. 1 點著色：相連的點塗上不同的顏色 3

Example 8. 2 4

Definition 8. 4 (著色數) 存在 3 -coloring c(G) 3 有子圖是K 3 不存在 2 -coloring c(G) 3 c(G) = 3 5

Homework Ex 8. 2 Determine the chromatic number of the Petersen graph. 6

Observation 8. 7 Theorem 8. 9 (by induction on k) 7

8. 2 Multipartite Graphs Example 8. 10 8

Homework 9

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Corollary 8. 11 Definition 8. 12 11

8. 3 Results for General Graphs Theorem 8. 19 12

Example G N 1={} (編號比自己小的鄰居) N 2={u 1} 1 u 1 N 3={} 2 2 u 4 N 4={u 1} u 2 u 5 3 u 6 u 3 1 D=3, 用 4色 N 5={u 2, u 3, u 4} N 6={u 2, u 3, u 4} 3 S 1=(1) (鄰居未使用的最小顏色) S 2=(1, 2) S 3=(1, 2, 1) S 4=(1, 2, 1, 2) S 5=(1, 2, 3) S 6=(1, 2, 3, 3) 13

Homework Ex: For the following graph G, find the colorings given by Delta Plus One Coloring Algorithm. What is the chromatic number c(G) of G? G u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 節點編號會影響顏色數 14

Example 8. 21 Theorem 8. 22 Definition 8. 24 15

Theorem 25 16

依照degree由大到小為節點編號 G N 2={u 1} 2 u 5 N 3={u 2} 1 1 u 1 N 4={u 1, u 3} u 3 u 4 2 u 2 N 1={} (編號比自己小的鄰居) N 5={u 1, u 3} N 6={u 2, u 4} 2 u 6 1 c(G) max(min{1, 4}, min{2, 4}, min{3, 4}, min{4, 4}, min{5, 3}, min{6, 3})=4 S 1=(1) (鄰居未使用的最小顏色) S 2=(1, 2) S 3=(1, 2, 1) S 4=(1, 2, 1, 2) S 5=(1, 2, 2) S 6=(1, 2, 2, 1) 只需2色!! 17

依照degree由大到小為節點編號 G 3 N 2={u 1} 3 u 5 N 3={u 1, u 2} u 2 2 1 u 4 u 3 N 1={} (編號比自己小的鄰居) N 4={u 1, u 2} N 5={u 1, u 2} N 6={u 3, u 4} 3 u 6 1 c(G) max(min{1, 5}, min{2, 5}, min{3, 4}, min{4, 4}, min{5, 3}, min{6, 3})=4 S 1=(1) (鄰居未使用的最小顏色) S 2=(1, 2) S 3=(1, 2, 3) S 4=(1, 2, 3, 3) S 5=(1, 2, 3, 3, 3) S 6=(1, 2, 3, 3, 3, 1) 只需3色 18

Homework Ex: For the following graph G, find the colorings given by Delta Plus One Coloring Algorithm and Greedy Coloring Algorithm. What is the chromatic number c(G) of G? u 1 G u 7 u 5 u 8 u 4 u 2 u 6 u 3 19

8. 4 Planar Graphs Theorem 8. 29 (Four Color) For a planar graph G, we have c(G) 4. 20

8. 5 Edge Coloring of Graphs Definition 8. 35 (幫edge著色，共用端點的edge必須使用不同顏色) Definition 8. 37 21

Observation 8. 40 K 4 u 1 3 1 2 2 u 4 u 1 u 5 u 2 u 4 1 3 u 3 Ex: 請給出一個 5 -edge coloring 22

Theorem 8. 41 K 3, 3 u 1 v 1 u 2 v 2 u 3 v 3 u 4 v 4 Corollary 8. 42 23

Theorem 8. 43 24