Colouring Complete Multipartite and Kneser-type Digraphs

TL;DR

This paper investigates the dichromatic number of various graphs, including Kneser and Borsuk graphs, extending classical results and exploring their list versions, with implications for graph coloring and product graphs.

Contribution

It extends classical results on the dichromatic number to Kneser and Borsuk graphs and analyzes their list versions, providing new bounds and directed analogues of known theorems.

Findings

Dichromatic number of Kneser graph $KG(n,k)$ is $ heta(n-2k+2)$.

List dichromatic number of $KG(n,k)$ is $ heta(n \\ln n)$ for certain parameters.

List dichromatic number of complete $r$-partite graphs is $ heta(r \\ln m)$.

Abstract

The dichromatic number of a digraph $D$ is the smallest $k$ such that $D$ can be partitioned into $k$ acyclic subdigraphs, and the dichromatic number of an undirected graph is the maximum dichromatic number over all its orientations. Extending a well-known result of Lov\'{a}sz, we show that the dichromatic number of the Kneser graph $KG(n,k)$ is $\Theta(n-2k+2)$ and that the dichromatic number of the Borsuk graph $BG(n+1,a)$ is $n+2$ if $a$ is large enough. We then study the list version of the dichromatic number. We show that, for any $\varepsilon>0$ and $2\leq k\leq n^{1/2-\varepsilon}$, the list dichromatic number of $KG(n,k)$ is $\Theta(n\ln n)$. This extends a recent result of Bulankina and Kupavskii on the list chromatic number of $KG(n,k)$, where the same behaviour was observed. We also show that for any $\rho>3$, $r\geq 2$ and $m\geq\max\{\ln^{\rho}r,2\}$, the list dichromatic number of the complete $r$-partite graph with $m$ vertices in each part is $\Theta(r\ln m)$, extending a classical result of Alon. Finally, we give a directed analogue of Sabidussi's theorem on the chromatic number of graph products.