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arXiv · 1910.07866

Edge-critical subgraphs of Schrijver graphs

Tomáš Kaiser·Matěj Stehlík
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Abstract

For $k\geq 1$ and $n\geq 2k$, the Kneser graph $KG(n,k)$ has all $k$-element subsets of an $n$-element set as vertices; two such subsets are adjacent if they are disjoint. It was first proved by Lov\'{a}sz that the chromatic number of $KG(n,k)$ is $n-2k+2$. Schrijver constructed a vertex-critical subgraph $SG(n,k)$ of $KG(n,k)$ with the same chromatic number. For the stronger notion of criticality defined in terms of removing edges, however, no analogous construction is known except in trivial cases. We provide such a construction for $k=2$ and arbitrary $n\geq 4$ by means of a nice explicit combinatorial definition.

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Tomáš Kaiser, Matěj Stehlík. 2019-10-17. Edge-critical subgraphs of Schrijver graphs. https://doi.org/10.1016/j.jctb.2020.02.004

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