论文标题
超图发射着色
Hypergraph incidence coloring
论文作者
论文摘要
HyperGraph $ \ MATHCAL {H} =（x，s）$的发病率是$（x，s）$，在x $中，$ x \ in s $ in s $，$ s \ in s $。如果（i）$ x = x'$或（ii）$ \ {x，x，x'\} \ subseteq s $或$ \ subseteq s $或$ \ {x，x，x，x，x'\} \ subSeteq s'$。适当的发病率$ k $ - 颜色的超毛图$ \ nathcal {h} $是从$ \ nathcal {h} $到$ \ \ \ {1,2，\ ldots，k \} $的映射$φ$的映射$φ$，因此$ \ Mathcal {H} $的$（x'，s'）$。 $ \ MATHCAL {H} $的发射率$χ_i（\ Mathcal {h}）$是最小整数$ k $，因此$ \ Mathcal {h} $具有适当的发病率$ k $ - 颜色。在本文中，我们证明了$χ_i（\ Mathcal {h}）\ leq（4/3+O（1））r（\ Mathcal {h}）δ（\ Mathcal {h}）$，每个$ t $ t $ t $ -quasi-linear Hypergraph带有$ t << r（ $δ（\ Mathcal {h}）$，其中$ r（\ Mathcal {h}）$是$ \ Mathcal {H} $中边缘的最大值。还证明，$χ_i（\ Mathcal {h}）\leqΔ（\ Mathcal {h}）+r（\ Mathcal {h}） - 1 $如果$ \ MATHCAL {H} $是$α$ -ASCALCAL {h} $是$α$ -ASCACY-acCyclic linear hypergraph，并且此限制为界面。
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.
