Star chromatic numbers of graphs

We investigate the relation between the star-chromatic number chi*(G) and the chromatic number chi(G) of a graph G. First we give a sufficient condition for graphs under which their starchromatic numbers are equal to their ordinary chromatic numbers. As a corollary we show that for any two positive integers k, g, there exists a R-chromatic graph of girth at least g whose starchromatic number is also k. The special case of this corollary with g = 4 answers a question of Abbott and Zhou. We also present an infinite family of triangle-free planar graphs whose starchromatic number equals their chromatic number. We then study the star-chromatic number of color-critical graphs. We prove that if an (m + 1)-critical graph has large girth, then its starchromatic number is close to m. We also consider (m + 1)-critical graphs with high connectivity. An infinite family of graphs is constructed to show that for each epsilon > 0 and each m greater than or equal to 2 there is an m-connected (m + 1)-critical graph with star chromatic number at most m + epsilon. This answers another question asked by Abbott and Zhou.