Holroyd, Fred and Talbot, John
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|DOI (Digital Object Identifier) Link:||http://doi.org/10.1016/j.disc.2004.08.028|
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For a graph G, vertex v of G and integer r >= 1, we denote the family of independent r-sets of V(G) by I^(r)(G) and the subfamily by I^(r)_v(G); such a family is called a star. Then, G is said to be r-EKR if no intersecting subfamily of I^(r)(G) is larger than the largest star in I^(r)(G). If every intersecting subfamily of I^(r)_v(G) of maximum size is a star, then G is said to be strictly r-EKR. We show that if a graph is r-EKR then its lexicographic product with any complete graph is r-EKR.
For any graph G, we define mu(G) to be the minimum size of a maximal independent vertex set. We conjecture that, if 1 <= r <= 1/2 mu(G), then G is r-EKR, and if r < 1/2 mu(G), then G is strictly r-EKR. This is known to be true when G is an empty graph, a cycle, a path or the disjoint union of complete graphs. We show that it is also true when G is the disjoint union of a pair of complete multipartite graphs.
|Item Type:||Journal Article|
|Extra Information:||Some of the symbols may not have transferred correctly into this bibliographic record and/or abstract.|
|Keywords:||Erdos-Ko-Rado theorem; EKR property; graphs; independent vertex sets|
|Academic Unit/Department:||Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
|Depositing User:||Fred Holroyd|
|Date Deposited:||27 Jun 2006|
|Last Modified:||03 Aug 2016 23:53|
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