For a graph $G=(V,E)$ and positive integer $p$, the exact distance-$p$ graph $G^{[\natural p]}$ is the graph with vertex set $V$ and with an edge between vertices $x$ and $y$ if and only if $x$ and $y$ have distance $p$. Recently, there has been an effort to obtain bounds on the chromatic number $\chi(G^{[\natural p]})$ of exact distance-$p$ graphs for $G$ from certain classes of graphs. In particular, if a graph $G$ has tree-width $t$, it has been shown that $\chi(G^{[\natural p]}) \in \mathcal{O}(p^{t-1})$ for odd $p$, and $\chi(G^{[\natural p]}) \in \mathcal{O}(p^{t}\Delta(G))$ for even $p$. We show that if $G$ is chordal and has tree-width $t$, then $\chi(G^{[\natural p]}) \in \mathcal{O}(p\, t^2)$ for odd $p$, and $\chi(G^{[\natural p]}) \in \mathcal{O}(p\, t^2 \Delta(G))$ for even $p$. If we could show that for every graph $H$ of tree-width $t$ there is a chordal graph $G$ of tree-width $t$ which contains $H$ as an isometric subgraph (i.e., a distance preserving subgraph), then our results would extend to all graphs of tree-width $t$. While we cannot do this, we show that for every graph $H$ of genus $g$ there is a graph $G$ which is a triangulation of genus $g$ and contains $H$ as an isometric subgraph.

中文翻译：

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为弦图的精确距离图着色

对于图 $G=(V,E)$ 和正整数 $p$，精确距离-$p$ 图 $G^{[\natural p]}$ 是顶点集 $V$ 和顶点 $x$ 和 $y$ 之间的边当且仅当 $x$ 和 $y$ 的距离为 $p$。最近，有人试图从某些类别的图中获得精确距离-$p$ 图的色数 $\chi(G^{[\natural p]})$ 的边界。特别地，如果图 $G$ 的树宽为 $t$，则表明 $\chi(G^{[\natural p]}) \in \mathcal{O}(p^{t-1 })$ 表示奇数 $p$，而 $\chi(G^{[\natural p]}) \in \mathcal{O}(p^{t}\Delta(G))$ 表示偶数 $p$。我们证明，如果 $G$ 是弦的并且具有树宽 $t$，那么 $\chi(G^{[\natural p]}) \in \mathcal{O}(p\, t^2)$ 对于奇数 $p$ 和 $\chi(G^{[\natural p]}) \in \mathcal{O}(p\, t^2 \Delta(G))$ 偶数 $p$。如果我们可以证明对于树宽 $t$ 的每个图 $H$ 都有一个树宽 $t$ 的弦图 $G$，其中包含 $H$ 作为等距子图（即，保持距离的子图） ，那么我们的结果将扩展到所有树宽 $t$ 的图。虽然我们不能这样做，但我们证明对于每个属 $g$ 的图 $H$ 都有一个图 $G$，它是属 $g$ 的三角剖分并且包含 $H$ 作为等距子图。