Hypergraph incidence coloring

Abstract

An incidence of a hypergraph H=(X,S) is a pair (x,s) with x∈X, s∈S, and x∈s. Two incidences (x,s) and (x′,s′) are adjacent if (i) x=x′, or (ii) {x,x′}⊆s or {x,x′}⊆s′. A proper incidence k-coloring of a hypergraph H is a mapping φ from the set of incidences of H to {1,2,…,k} so that φ(x,s)≠φ(x′,s′) for every two adjacent incidences (x,s) and (x′,s′) of H. The incidence chromatic number χI(H) of H is the minimum integer k such that H has a proper incidence k-coloring. In this paper we prove χI(H)≤(4/3+o(1))r(H)Δ(H) for every t-quasi-linear hypergraph with t<<r(H) and with sufficiently large maximum degree Δ(H), where r(H) is the maximum of the cardinalities of the edges in H. It is also proved that χI(H)≤Δ(H)+r(H)−1 if H is an α-acyclic linear hypergraph, and this bound is sharp.