On multichromatic numbers of widely colorable graphs

Abstract

AbstractA coloring is called ‐wide if no walk of length connects vertices of the same color. A graph is ‐widely colorable with colors if and only if it admits a homomorphism into a universal graph . Tardif observed that the value of the multichromatic number of these graphs is at least and equality holds for . He asked whether there is equality also for . We show that for all thereby answering Tardif's question. We observe that for large (with respect to and fixed) we cannot have equality and that for fixed and going to infinity the fractional chromatic number of also tends to infinity. The latter is a simple consequence of another result of Tardif on the fractional chromatic number of generalized Mycielski graphs.