Chromatic Numbers of Exact Distance Graphs

TL;DR

This paper provides simplified proofs and improved bounds for the chromatic number of exact distance graphs, showing it is controlled by generalized coloring numbers for various graph classes.

Contribution

It offers a simpler proof for existing bounds and introduces tighter bounds on the chromatic number of exact distance graphs using generalized coloring numbers.

Findings

Chromatic number of odd p-distance graphs bounded by weak (2p-1)-coloring number.

For even p, the chromatic number is bounded by weak (2p)-coloring number times maximum degree.

Improved lower bounds for planar and $K_t$-minor free graphs.

Abstract

For any graph $G=(V,E)$ and positive integer $p$, the exact distance-$p$ graph $G^{[\natural p]}$ is the graph with vertex set $V$, which has an edge between vertices $x$ and $y$ if and only if $x$ and $y$ have distance $p$ in $G$. For odd $p$, Ne\v{s}et\v{r}il and Ossona de Mendez proved that for any fixed graph class with bounded expansion, the chromatic number of $G^{[\natural p]}$ is bounded by an absolute constant. Using the notion of generalised colouring numbers, we give a much simpler proof for the result of Ne\v{s}et\v{r}il and Ossona de Mendez, which at the same time gives significantly better bounds. In particular, we show that for any graph $G$ and odd positive integer $p$, the chromatic number of $G^{[\natural p]}$ is bounded by the weak $(2p-1)$-colouring number of $G$. For even $p$, we prove that $\chi(G^{[\natural p]})$ is at most the weak $(2p)$-colouring number times the maximum degree. For odd $p$, the existing lower bound on the number of colours needed to colour $G^{[\natural p]}$ when $G$ is planar is improved. Similar lower bounds are given for $K_t$-minor free graphs.