The asymptotic behavior of the correspondence chromatic number

Alon (2000) proved that for any graph G , ź ź ( G ) = ź ( ln d ) , where ź ź ( G ) is the list chromatic number of G and d is the average degree of G . Dvořak 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 ź c ( G ) = ź ( d / ln d ) , where ź c ( G ) denotes the correspondence chromatic number of G . We also prove that for triangle-free G , ź c ( G ) = O ( Δ / ln Δ ) , 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.