DP-colorings of hypergraphs

Anton Bernshteyn, Alexandr Kostochka

Research output: Contribution to journalArticlepeer-review

Abstract

Classical problems in hypergraph coloring theory are to estimate the minimum number of edges, m 2 (r) (respectively, m 2 (r)), in a non-2-colorable r-uniform (respectively, r-uniform and simple) hypergraph. The best currently known bounds are c⋅r∕logr⋅2 r ⩽m 2 (r)⩽C⋅r 2 ⋅2 r andc ⋅r −ε ⋅4 r ⩽m 2 (r)⩽C ⋅r 4 ⋅4 r ,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)⩽m 2 (r) and m˜ 2 (r)⩽m 2 (r). While the proof of the bound m 2 (r)=Ω(r −3 4 r ) due to Erdős and Lovász also works for m˜ 2 (r), we show that the trivial lower bound m˜ 2 (r)⩾2 r−1 is asymptotically tight, i.e., m˜ 2 (r)⩽(1+o(1))2 r−1 . On the other hand, when r⩾2 is even, we prove that the lower bound m˜ 2 (r)⩾2 r−1 is not sharp, i.e., m˜ 2 (r)⩾2 r−1 +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)−2 r−1 can be arbitrarily large.

Original languageEnglish (US)
Pages (from-to)134-146
Number of pages13
JournalEuropean Journal of Combinatorics
Volume78
DOIs
StatePublished - May 2019

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

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