# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2018-2019): c. 84%

# Discussiones Mathematicae Graph Theory

Article in press

Authors:

T. Schweser

Title:

Generalized hypergraph coloring

Source:

Discussiones Mathematicae Graph Theory

Received: 2018-04-18, Revised: 2018-08-23, Accepted: 2018-08-23, https://doi.org/10.7151/dmgt.2168

Abstract:

A smooth hypergraph property $\mathcal{P}$ is a class of hypergraphs that is hereditary and non-trivial, i.e., closed under induced subhypergraphs and it contains a non-empty hypergraph but not all hypergraphs. In this paper we examine $\mathcal{P}$-colorings of hypergraphs with smooth hypergraph properties $\mathcal{P}$. A $\mathcal{P}$-coloring of a hypergraph $H$ with color set $C$ is a function $\varphi:V(H) \to C$ such that $H[\varphi^{-1}(c)]$ belongs to $\mathcal{P}$ for all $c \in C$. Let $L: V(H) \to 2^C$ be a so called list-assignment of the hypergraph $H$. Then, a $(\mathcal{P},L)$-coloring of $H$ is a $\mathcal{P}$-coloring $\varphi$ of $H$ such that $\varphi(v) \in L(v)$ for all $v \in V(H)$. The aim of this paper is a characterization of $(\mathcal{P},L)$-critical hypergraphs. Those are hypergraphs $H$ such {that} $H-v$ is $(\mathcal{P},L)$-colorable for all $v \in V(H)$ but $H$ itself is not. Our main theorem is a Gallai-type result for critical hypergraphs, which implies a Brooks-type result for $(\mathcal{P},L)$-colorable hypergraphs. In the last section, we prove a Gallai-type bound for the degree sum of $(\mathcal{P},L)$-critical locally simple hypergraphs.

Keywords:

hypergraph decomposition, vertex partition, degeneracy, coloring of hypergraphs, hypergraph properties