A note on the Berge–Meyniel conjecture

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.