Colouring Complete Multipartite and Kneser-type Digraphs
Ararat Harutyunyan, Gil Puig i Surroca

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 is the smallest such that can be partitioned into 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 is and that the dichromatic number of the Borsuk graph is if is large enough. We then study the list version of the dichromatic number. We show that, for any and , the list dichromatic number of is . This extends a recent result of Bulankina and Kupavskii on the list chromatic number of , where the same behaviour was observed. We also show that for any , and , the list dichromatic…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Combinatorial Mathematics · Advanced Graph Theory Research
