Journal of Mathematical Research & Exposition Sept., 2010, Vol. 30, No. 5, pp. 841–844 DOI:10.3770/j.issn:1000-341X.2010.05.009 Http://jmre.dlut.edu.cn A Note on Star Chromatic Number of Graphs Hong Yong FU 1,2, * , De Zheng XIE 1 1. College of Mathematics and Statistics, Chongqing University, Chongqing 401331, P. R. China; 2. College of Economics and Business Administration, Chongqing University, Chongqing 400044, P. R. China Abstract A star coloring of an undirected graph G is a proper coloring of G such that no path of length 3 in G is bicolored. The star chromatic number of an undirected graph G, denoted by χs(G), is the smallest integer k for which G admits a star coloring with k colors. In this paper, we show that if G is a graph with maximum degree Δ, then χs(G) ≤⌈7Δ 3 2 ⌉, which gets better bound than those of Fertin, Raspaud and Reed. Keywords star coloring; star chromatic number; proper coloring. Document code A MR(2000) Subject Classification 05C15; 05C80 Chinese Library Classification O157.5 1. Introduction All graphs considered here are undirected graphs. A star coloring of an undirected graph G is a proper coloring (i.e., no two neighbors are assigned the same color) of G such that any path of length 3 in G is not bicolored. The star chromatic number of undirected graph G, denoted by χ s (G), is the smallest integer k for which G admits a star coloring with k colors. The terminologies and notations used but undefined in this paper can be found in [2, 3]. Star coloring was introduced in 1973 by Gr¨ unbaume [4]. In 2001, Nesetril et al. [5] proved that χ s (G) ≤ O(Δ 2 ). In 2004, Albertson et al. [6] proved that χ s (G) ≤ Δ(Δ − 1) + 2. In 2004, Fertin et al. [1] proved that χ s (G) ≤⌈20Δ 3 2 ⌉. In this paper, we extend those results above and give a good bound for χ s (G), i.e., we show that if G is a graph with maximum degree Δ, then χ s (G) ≤⌈7Δ 3 2 ⌉, which is better than any of the above bounds. 2. Lemmas and the main result Let G * be a graph with vertex set X. We say that G * on the set X is a dependency graph for the family of event (A x ) x∈X (i.e., any two events A x and A y (x, y ∈ X ) will share an edge in G * iff they are dependent). Erd¨ os and Lov´ asz [7] proved the following fundamental lemma, namely, Received October 25, 2008; Accepted May 16, 2009 Supported by the Natural Science Foundation of Chongqing Science and Technology Commission (Grant No. 2007BB2123). * Corresponding author E-mail address: fuhongyong2007@163.com (H. Y. FU); xdz@cqu.edu.cn (D. Z. XIE)