On multichromatic numbers of widely colorable graphs
Abstract
A coloring is called s-wide if no walk of length 2s-1 connects vertices of the same color. A graph is s-widely colorable with t colors if and only if it admits a homomorphism into a universal graph W(s,t). Tardif observed that the value of the r th multichromatic number r(W(s,t)) of these graphs is at least t+2(r-1) and equality holds for r=s=2. He asked whether there is equality also for r=s=3. We show that s(W(s,t))=t+2(s-1) for all s thereby answering Tardif's question. We observe that for large r (with respect to s and t fixed) we cannot have equality and that for s fixed and t going to infinity the fractional chromatic number of W(s,t) also tends to infinity. The latter is a simple consequence of another result of Tardif on the fractional chromatic number of generalized Mycielski graphs.