34 research outputs found
Dynamic Chromatic Number of Regular Graphs
A dynamic coloring of a graph is a proper coloring such that for every
vertex of degree at least 2, the neighbors of receive at least
2 colors. It was conjectured [B. Montgomery. {\em Dynamic coloring of graphs}.
PhD thesis, West Virginia University, 2001.] that if is a -regular
graph, then . In this paper, we prove that if is a
-regular graph with , then . It confirms the conjecture for all regular graph with
diameter at most 2 and . In fact, it shows that
provided that has diameter at most 2 and
. Moreover, we show that for any -regular graph ,
. Also, we show that for any there exists a
regular graph whose chromatic number is and .
This result gives a negative answer to a conjecture of [A. Ahadi, S. Akbari, A.
Dehghan, and M. Ghanbari. \newblock On the difference between chromatic number
and dynamic chromatic number of graphs. \newblock {\em Discrete Math.}, In
press].Comment: 8 page