DP-colorings of hypergraphs

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Abstract

Classical problems in hypergraph coloring theory are to estimate the minimum number of edges, m2(r) (respectively, m2(r)), in a non-2-colorable r-uniform (respectively, r-uniform and simple) hypergraph. The best currently known bounds are crlogr2rm2(r)Cr22randcrε4rm2(r)Cr44r,for any fixed ε>0 and some c, c, C, C>0 (where c may depend on ε). In this paper we consider the same problems in the context of DP-coloring (also known as correspondence coloring), which is a generalization of list coloring introduced by Dvořák and Postle and related to local conflict coloring studied independently by Fraigniaud, Heinrich, and Kosowski. Let m˜2(r) (respectively, m˜2(r)) denote the minimum number of edges in a non-2-DP-colorable r-uniform (respectively, r-uniform and simple) hypergraph. By definition, m˜2(r)m2(r) and m˜2(r)m2(r).

While the proof of the bound m2(r)=Ω(r34r) due to Erdős and Lovász also works for m˜2(r), we show that the trivial lower bound m˜2(r)2r1 is asymptotically tight, i.e., m˜2(r)(1+o(1))2r1. On the other hand, when r2 is even, we prove that the lower bound m˜2(r)2r1 is not sharp, i.e., m˜2(r)2r1+1. Whether this result holds for infinitely many odd values of r remains an open problem. Nevertheless, we conjecture that the difference m˜2(r)2r1 can be arbitrarily large.

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Research of the first author is supported in part by the Waldemar J., Barbara G., and Juliette Alexandra Trjitzinsky Fellowship, USA. Research of the second author is supported in part by NSF, USA grant DMS-1600592 and grants 18-01-00353A and 16-01-00499 of the Russian Foundation for Basic Research .