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 any two adjacent incidences (x, s) and (x , s ) of H. The incidence chromatic number χ I (H) of H is the minimum integer k such that H has a proper incidence k-coloring.In this paper we prove χ I (H) ≤ (4/3 + o(1))r(H)∆(H) for every t-quasi-linear hypergraph with t << r(H) and sufficiently large ∆(H), where r(H) is the maximum of the cardinalities of the edges in H.It is also proved that χ I (H) ≤ ∆(H) + r(H) − 1 if H is an α-acyclic linear hypergraph, and this bound is sharp.