Multi-Colouring of Kneser Graphs: Notes on Stahl's Conjecture

TL;DR

This paper investigates multi-colourings of Kneser graphs, addressing Stahl's conjecture by providing new results, simplified proofs, and extending understanding of when certain graphs are colourable with different parameters.

Contribution

It offers new insights and simplified proofs related to Stahl's conjecture on multi-colourings of Kneser graphs, advancing the theoretical understanding of graph colourability.

Findings

New results on Stahl's conjecture

Simplified proofs of known cases

Extended conditions for multi-colourability

Abstract

A (finite, undirected) graph is $(n,k)$-colourable if we can assign each vertex a $k$-subset of $\{1,2,\ldots,n\}$ so that adjacent vertices receive disjoint subsets. We consider the following problem: if a graph is $(n,k)$-colourable, then for what pairs $(n',k')$ is it also $(n',k')$-colourable? This question can be translated into a question regarding multi-colourings of Kneser graphs, for which Stahl formulated a conjecture in 1976. We present new results, strengthen existing results, and in particular present much simpler proofs of several known cases of the conjecture.