1. On the Dynamic Colorings of Graphs
陳伯亮 台中技術學院
黃國卿 靜宜大學

2. Defintion.
A dynamic coloring of a graph G is a proper coloring such that for every vertex v  V(G) of degree at least 2, the neighbors of v receive at least 2 colors.
The smallest integer k that there is a dynamic k-coloring of G is called the dynamic chromatic number of G and denoted by d(G).

3. Examples 1 3 2 C6 2 3 1 3

4. Examples 2 1 C4 3 4

5. Examples 1 2 3 C5 5

6. d(Cn)
Theorem.

7. Well-know Results Lemma. （Montgomery, 2001）
Let G be a connected, nontrivial graph. Then d(K2) = 2 and d(G)  3 otherwise.
Lemma.
If G is dynamic k-colorable, then G is dynamic (k+1)-colorable.

8. Example. d(C5) = 5, d(W5) = 4. 1 2 3 W5 4 2 1

9. Theorem.
For any graph G, (G)  (G)+1.
Theorem. （Montgomery, 2001）
For any graph G, d(G)  (G)+3, and equality holds only for G = C5.
Theorem. （Lai, Montgomery and Poon, 2003）
d(G)  (G)+1 if (G)  3.

10. Example. （Montgomery, 2001）
G(X,Y) be a bipartite graph, X=[n],
Then

11. 1 2 3 4 G 12 13 14 23 24 34

12. Conjecture. （Montgomery, 2001）
For any regular graph G, d(G)  (G)+2.

13. Definition.
The Kneser graphs KG(n,k) has the vertex set of all k-subsets of the n-set. Two vertices are different and to be adjacent in KG(n,k) if they have empty of intersection as subsets.

14. Definition.
The Kneser graphs KG(n,k) has the vertex set of all k-subsets of the n-set. Two vertices are different and to be adjacent in KG(n,k) if they have empty of intersection as subsets.
[n] ={kN : 1  k  n}

15. Theorem (Lovász, 1978)
(KG(n,k)) = n2k+2.

16. (KG(n,k))
for all 1  i  n-2k+1.

17. d(KG(n,k))
for all 1  i  n-2k+1.
A=[k]={1,2,…,k}
C(N(A))={n-2k+2}

18. d(KG(5,2)) 18

19. d(KG(5,2)) 19

20. d(KG(5,2))>  (KG(5,2)) 20

21. d(KG(n,k))
Theorem.
d(KG(n,k)) = (KG(n,k)) = n2k if n  2k+2 or k is odd.

22. d(KG(n,k))
Case A. n  2k+2
for all 1  i  n-2k.
where

23. d(KG(n,k))
Case B. n = 2k+1 and k is odd

24. d(KG(n,k))
Theorem.
d(KG(n,k)) = (KG(n,k)) = n2k if n  2k+2 or k is odd.

25. d(KG(n,k)) when k is even
Lemma.
d(KG(5,2)) = (KG(5,2))+1 = 4.
d(KG(9,4)) = (KG(9,4)) = 3.

26. Reference
B. Montgomery, Dynamic coloring of graphs, Ph.D. dissertation. West Virginia University, 2001.
H.J. Lai, B. Montgomery and H. Poon, Upper bounds of dynamic chromatic number, Ars Combin. 68(2003), 193–201.

27. 謝謝