Colouring Complete Multipartite and Kneser-type Digraphs

TL;DR

This work extends Lovász's paradigm from chromatic numbers to dichromatic numbers in digraphs, establishing that $\\vec{\\chi}(KG(n,k))=\\Theta(n-2k+2)$ and $\\vec{\\chi}(BG(n+1,a))=n+2$ for large $a$, thereby linking topological methods with digraph coloring. It then develops the list-dichromatic theory, proving that $\\vec{\\chi}_{\\ell}(KG(n,k))=\\Theta(n\log n)$ for $2\le k\le n^{1/2-\\varepsilon}$ and that $\\vec{\\chi}_{\\ell}(K_{m*r})=\\Theta(r\log m)$ under suitable growth, aligning these values with known list-chromatic behavior. The paper also extends Sabidussi’s product theorem to digraphs, showing $\\vec{\\chi}(G\\square H)=\\max\\{\\vec{\\chi}(G),\\vec{\\chi}(H)\\}$, and discusses the relation to current results and open questions in directed coloring. Overall, it blends probabilistic, geometric, and topological techniques to transfer key undirected coloring phenomena to the directed and list-settings, with implications for understanding dichromatic structure in Kneser, Borsuk, and multipartite graphs.

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 Lovász, we show that the dichromatic number of the Kneser graph $KG(n,k)$ is $Θ(n-2k+2)$ 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 $Θ(n\ln n)$. 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 $ρ>3$, $r\geq 2$ and $m\geq\max\{\ln^ρr,2\}$, the list dichromatic number of the complete $r$-partite graph with $m$ vertices in each part is $Θ(r\ln m)$, extending a classical result of Alon. Finally, we give a directed analogue of Sabidussi's theorem on the chromatic number of graph products.

Theorems & Definitions (36)

• Conjecture 1.1
• Theorem 2.1
• Theorem 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• proof
• Theorem 2.6
• proof