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Edge-critical subgraphs of Schrijver graphs
Journal of Combinatorial Theory Series B ( IF 1.4 ) Pub Date : 2020-02-14 , DOI: 10.1016/j.jctb.2020.02.004
Tomáš Kaiser , Matěj Stehlík

For k1 and n2k, 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 Lovász that the chromatic number of KG(n,k) is n2k+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 n4 by means of a simple explicit combinatorial definition.



中文翻译:

Schrijver图的边缘关键子图

对于 ķ1个ñ2ķ,Kneser图 ķGñķn个元素的所有k个元素子集设置为顶点;如果两个这样的子集不相交,则它们是相邻的。洛瓦兹(Lovász)首先证明,ķGñķñ-2ķ+2。Schrijver构造了一个顶点关键子图小号GñķķGñķ具有相同的色数。然而,对于在去除边缘方面定义的更严格的临界概念,除了琐碎的情况之外,没有类似的构造是已知的。我们为ķ=2 和任意 ñ4 通过简单的显式组合定义。

更新日期:2020-02-14
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