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

TL;DR

The paper studies multi-colourings of Kneser graphs $K(n,k)$ and investigates Stahl's conjecture, which predicts $\chi_{k'}(K(n,k)) = qn - 2r$ for $k' = qk - r$ with $q\ge1$ and $0\le r\le k-1$. It derives a universal lower bound $\chi_{k'}(K(n,k)) \ge \lceil k'n/k \rceil$ via independence numbers and introduces a splitting method that partitions $[n]$ into blocks to obtain stronger bounds. Using these ideas, the authors prove Stahl's conjecture for the range $0 \le r \le k/(n-2k)$ (Theorem thm:main1) and establish a finite-n reduction (Theorem thm:main2) showing that verifying a finite set of cases suffices to settle the conjecture for all $n$ and $k$. They also provide simpler proofs for several known cases (notably $k=2,3$) and discuss the structural role of independent sets (via Erdős–Ko–Rado and Hilton–Milner bounds) in driving the lower bounds. The work narrows the conjecture to a finite verification set and highlights the remaining open cases (e.g., whether $\chi_{45}(K(11,4))=126$), while also addressing a symmetric question on mutual colourability between pairs $(n,k)$ and $(n',k')$.

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.

Theorems & Definitions (28)

• Conjecture 1.3: Stahl stahl1976n
• proof
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1: Geller and Stahl geller1975chromatic
• Lemma 2.2: Stahl stahl1976n
• Theorem 2.3
• Theorem 2.4