Star coloring of graphs

Abstract

A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two neighbors are assigned the same color) such that any path of length 3 in G is not 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 give the exact value of the star chromatic number of different families of graphs such as trees, cycles, complete bipartite graphs, outerplanar graphs, and 2-dimensional grids. We also study and give bounds for the star chromatic number of other families of graphs, such as planar graphs, hypercubes, d-dimensional grids (d ≥ 3), d-dimensional tori (d ≥ 2), graphs with bounded treewidth, and cubic graphs. We end this study by two asymptotic results, where we prove that, when d tends to infinity, (i) there exist graphs G of maximum degree d such that $\chi _s(G) = \Omega({d^{{3\over 2}}\backslash ({\rm log}\ {d})^{1\over 2}})$ and (ii) for any graph G of maximum degree d, $\chi _s(G) = O({d^{{3\over 2}}})$. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 163–182, 2004