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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
中文翻译:
Schrijver图的边缘关键子图
更新日期:2020-02-14
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 and , the Kneser graph 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 is . Schrijver constructed a vertex-critical subgraph of 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 and arbitrary by means of a simple explicit combinatorial definition.
中文翻译:
Schrijver图的边缘关键子图
对于 和 ,Kneser图 将n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, 是 。Schrijver构造了一个顶点关键子图 的 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 和任意 通过简单的显式组合定义。