A tight bound on the set chromatic number

Abstract

We provide a tight bound on the set chromatic number of a graph in terms of its chromatic number. Namely, for all graphs G, we show that χs(G) < ⌋log2 χ(G)⌉ + 1, where χs(G) and χ(G) are the set chromatic number and the chromatic number of G, respectively. This answers in the affirmative a conjecture of Gera, Okamoto, Rasmussen and Zhang.