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arXiv · 2407.05730

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

Jan van den Heuvel·Xinyi Xu
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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.

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Jan van den Heuvel, Xinyi Xu. 2024-07-08. Multi-Colouring of Kneser Graphs: Notes on Stahl's Conjecture. https://arxiv.org/abs/2407.05730

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