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 $\varphi$ from the set of incidences of $\mathcal{ H}$ to $\{ 1,2,\ldots,k\}$ so that $\varphi(x,s)\neq \varphi(x',s')$ for any two adjacent incidences $(x,s)$ and $(x',s')$ of $\mathcal{ H}$. The incidence chromatic number $\chi_I(\mathcal{ H})$ of $\mathcal{ H}$ is the minimum integer $k$ such that $\mathcal{ H}$ has a proper incidence $k$-coloring. In this paper we prove $\chi_I(\mathcal{ H})\leq (4/3+o(1))r(\mathcal{ H})\Delta(\mathcal{ H})$ for every $t$-quasi-linear hypergraph with $t<<r(\mathcal{ H})$ and sufficiently large $\Delta(\mathcal{ H})$, where $r(\mathcal{ H})$ is the maximum of the cardinalities of the edges in $\mathcal{ H}$. It is also proved that $\chi_I(\mathcal{ H})\leq \Delta(\mathcal{ H})+r(\mathcal{ H})-1$ if $\mathcal{ H}$ is an $\alpha$-acyclic linear hypergraph, and this bound is sharp.
Categories: math.CO