Coloring locally sparse graphs

Paper

Abstract

A graph

-locally sparse if for each vertex

, the subgraph induced by its neighborhood contains at most

edges. Alon, Krivelevich, and Sudakov showed that for

if a graph

of maximum degree

-locally-sparse, then

. We introduce a more general notion of local sparsity by defining graphs

to be

-locally-sparse for some graph

if for each vertex

the subgraph induced by the neighborhood of

contains at most

copies of

. Employing the Rödl nibble method, we prove the following generalization of the above result: for every bipartite graph

, if

-locally-sparse, then

. This improves upon results of Davies, Kang, Pirot, and Sereni who consider the case when

is a path. Our results also recover the best known bound on

when

-free for

, and hold for list and correspondence coloring in the more general so-called ''color-degree'' setting.