The asymptotic behavior of the correspondence chromatic number

Abstract

Alon (2000) proved that for any graph G, χℓ(G)=Ω(lnd), where χℓ(G) is the list chromatic number of G and d is the average degree of G. Dvořák and Postle (2015) recently introduced a generalization of list coloring, which they called correspondence coloring. We establish an analog of Alon’s result for correspondence coloring; namely, we show that χc(G)=Ω(d/lnd), where χc(G) denotes the correspondence chromatic number of G. We also prove that for triangle-free G, χc(G)=O(Δ/lnΔ), where Δ is the maximum degree of G (this is a generalization of Johansson’s result about list colorings (Johansson, 1996)). This implies that the correspondence chromatic number of a regular triangle-free graph is, up to a constant factor, determined by its degree.