A note on the Berge–Meyniel conjecture

Abstract

An edge coloring of hypergraph [Formula: see text] is a color assignment to all edges in [Formula: see text] such that any pair of intersecting edges receive different colors. Let [Formula: see text] denote the minimum number of colors among all edge colorings of [Formula: see text]. In 1985, Berge and Meyniel, independently, conjectured that each loopless linear hypergraph [Formula: see text] has [Formula: see text], where [Formula: see text]. In this paper, a description of chromatic critical hypergraphs is given. Furthermore, we show [Formula: see text] for [Formula: see text]-vertex-restricted hypergraphs and [Formula: see text] for edge-restricted hypergraphs. Since either of the class of 1-vertex-restricted hypergraphs and the class of edge-restricted hypergraphs contains well-ordered hypergraphs as a subclass, our results strictly extend the one on well-ordered hypergraphs due to Bretto et al. in 2020.