Star chromatic numbers of graphs

We investigate the relation between the star-chromatic number ?(G) and the chromatic number ?(G) of a graphG. 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 integersk, g, there exists ak-chromatic graph of girth at leastg whose star-chromatic number is alsok. The special case of this corollary withg=4 answers a question of Abbott and Zhou. We also present an infinite family of triangle-free planar graphs whose star-chromatic number equals their chromatic number. We then study the star-chromatic number of An infinite family of graphs is constructed to show that for each e>0 and eachm=2 there is anm-connected (m+1)-critical graph with star chromatic number at mostm+e. This answers another question asked by Abbott and Zhou.