DP-Colorings of Hypergraphs

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, $m_2(r)$ (respectively, $m^\ast_2(r)$), in a non-$2$-colorable $r$-uniform (respectively, $r$-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 $\varepsilon > 0$ and some $c$, $c'$, $C$, $C' > 0$ (where $c'$ may depend on $\varepsilon$). 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 $\tilde{m}_2(r)$ (respectively, $\tilde{m}^\ast_2(r)$) denote the minimum number of edges in a non-$2$-DP-colorable $r$-uniform (respectively, $r$-uniform and simple) hypergraph. By definition, $\tilde{m}_2(r) \leqslant m_2(r)$ and $\tilde{m}^\ast_2(r)\leqslant m^\ast_2(r)$. While the proof of the bound $m^\ast_2(r) = \Omega( r^{-3} 4^r)$ due to Erd\H{o}s and Lov\'{a}sz also works for $\tilde{m}^\ast_2(r)$, we show that the trivial lower bound $\tilde{m}_2(r) \geqslant 2^{r-1}$ is asymptotically tight, i.e., $\tilde{m}_2(r) \leqslant (1 + o(1))2^{r-1}$. On the other hand, when $r \geqslant 2$ is even, we prove that the lower bound $\tilde{m}_2(r) \geqslant 2^{r-1}$ is not sharp, i.e., $\tilde{m}_2(r) \geqslant 2^{r-1}+1$. Whether this result holds for any odd values of $r$ remains an open problem. Nevertheless, we conjecture that the difference $\tilde{m}_2(r) - 2^{r-1}$ can be arbitrarily large.