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arXiv · 2412.20001

Colouring signed analogues of Kneser, Schrijver, and Borsuk graphs

Luis Kuffner·Reza Naserasr·Lujia Wang·Xiaowei Yu·Huan Zhou·Xuding Zhu
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Abstract

The Kneser signed graph $\KS(n,k)$, $k\leq n$, is the graph whose vertices are signed $k$-subsets of $[n]$ (i.e. $k$-subsets $S$ of $\{ \pm 1, \pm 2, \ldots, \pm n\}$ such that $S\cap (-S)=\emptyset$). Two vertices $A$ and $B$ are adjacent with a positive edge if $A\cap (-B)=\emptyset$ and with a negative edge if $A\cap B=\emptyset$. We prove that the balanced chromatic number of $\KS(n,k)$ is $n-k+1$. We then introduce the signed analogue of Schrijver graphs and show that they form vertex-critical subgraphs of $\KS(n,k)$ with respect to balanced colouring. Further connection to topological methods, in particular, connection to Borsuk signed graphs is also considered.

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Luis Kuffner, Reza Naserasr, Lujia Wang, Xiaowei Yu, Huan Zhou, Xuding Zhu. 2024-12-28. Colouring signed analogues of Kneser, Schrijver, and Borsuk graphs. https://arxiv.org/abs/2412.20001

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