Hypergraph incidence coloring

An incidence of a hypergraph $\mathcal{H}=(X,S)$ is a pair $(x,s)$ with $x\in X$, $s\in S$ and $x\in s$. Two incidences $(x,s)$ and $(x',s')$ are adjacent if (i) $x=x'$, or (ii) $\{x,x'\}\subseteq s$ or $\{x,x'\}\subseteq s'$. A proper incidence $k$-coloring of a hypergraph $\mathcal{H}$ is a mapping $φ$ from the set of incidences of $\mathcal{H}$ to $\{1,2,\ldots,k\}$ so that $φ(x,s)\neq φ(x',s')$ for any two adjacent incidences $(x,s)$ and $(x',s')$ of $\mathcal{H}$. The incidence chromatic number $χ_I(\mathcal{H})$ of $\mathcal{H}$ is the minimum integer $k$ such that $\mathcal{H}$ has a proper incidence $k$-coloring. In this paper we prove $χ_I(\mathcal{H})\leq (4/3+o(1))r(\mathcal{H})Δ(\mathcal{H})$ for every $t$-quasi-linear hypergraph with $t<<r(\mathcal{H})$ and sufficiently large $Δ(\mathcal{H})$, where $r(\mathcal{H})$ is the maximum of the cardinalities of the edges in $\mathcal{H}$. It is also proved that $χ_I(\mathcal{H})\leq Δ(\mathcal{H})+r(\mathcal{H})-1$ if $\mathcal{H}$ is an $α$-acyclic linear hypergraph, and this bound is sharp.

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