DP-Colorings of Hypergraphs
Anton Bernshteyn, Alexandr Kostochka

TL;DR
This paper investigates DP-coloring in hypergraphs, establishing bounds on the minimum edges needed for non-2-DP-colorable hypergraphs and revealing asymptotic tightness of trivial bounds.
Contribution
It extends classical hypergraph coloring bounds to the DP-coloring context, providing new asymptotic results and conjectures on the minimal edges for non-2-DP-colorable hypergraphs.
Findings
Trivial lower bound for {m}~_2(r) is asymptotically tight.
For even r {m}~_2(r) {ge} 2^{r-1}+1.
Open problem on the behavior for odd r.
Abstract
Classical problems in hypergraph coloring theory are to estimate the minimum number of edges, (respectively, ), in a non--colorable -uniform (respectively, -uniform and simple) hypergraph. The best currently known bounds are \[c \cdot \sqrt{r/\log r} \cdot 2^r \,\leqslant\, m_2(r) \,\leqslant\, C \cdot r^2 \cdot 2^r \qquad \text{and} \qquad c' \cdot r^{-\varepsilon} \cdot 4^r \,\leqslant\, m_2^\ast(r) \,\leqslant\, C' \cdot r^4 \cdot 4^r,\] for any fixed and some , , , (where may depend on ). 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 Dvo\v{r}\'{a}k and Postle and related to local conflict coloring studied independently by Fraigniaud, Heinrich, and Kosowski. Let…
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