论文标题
1平面图的强色指数
The strong chromatic index of 1-planar graphs
论文作者
论文摘要
图$ g $的色度索引$χ'（g）$是$ g $的最小$ k $，$ g $承认边缘$ k $ - 颜色，以使任何两个相邻的边缘具有不同的颜色。 $ g $的强色索引$χ'_s（g）$是最小的$ k $，因此$ g $具有边缘$ k $ - 颜色的，条件是距离最多两个边的任何两个边都会获得不同的颜色。如果可以在平面中绘制图形，则图为1平面，使每个边缘在最多另一个边缘越过。在本文中，我们表明每个图$ g $具有最高平均度$ \ bar {d}（g）$具有$χ'__ {s}（g）（g）\ le（2 \ bar {d}（g）-1 -1）χ'（g）$。作为推论，我们证明，每1平面图$ g $具有最高度$δ$具有$χ'_{\ rm s}（g）\ le14δ$，这会改善结果，这是由于Bensmail等人的，这说明$χ'__ _ _ _ _ {\ rm s}（g）（g）（g）\ le24δ$
The chromatic index $χ'(G)$ of a graph $G$ is the smallest $k$ for which $G$ admits an edge $k$-coloring such that any two adjacent edges have distinct colors. The strong chromatic index $χ'_s(G)$ of $G$ is the smallest $k$ such that $G$ has an edge $k$-coloring with the condition that any two edges at distance at most 2 receive distinct colors. A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, we show that every graph $G$ with maximum average degree $\bar{d}(G)$ has $χ'_{s}(G)\le (2\bar{d}(G)-1)χ'(G)$. As a corollary, we prove that every 1-planar graph $G$ with maximum degree $Δ$ has $χ'_{\rm s}(G)\le 14Δ$, which improves a result, due to Bensmail et al., which says that $χ'_{\rm s}(G)\le 24Δ$ if $Δ\ge 56$.
