Holroyd, Fred and Talbot, John
(2005).

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DOI (Digital Object Identifier) Link:  https://doi.org/10.1016/j.disc.2004.08.028 

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Abstract
For a graph G, vertex v of G and integer r >= 1, we denote the family of independent rsets 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 rEKR 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 rEKR. We show that if a graph is rEKR then its lexicographic product with any complete graph is rEKR.
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 rEKR, and if r < 1/2 mu(G), then G is strictly rEKR. 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 Item 

ISSN:  0012365X 
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Keywords:  ErdosKoRado theorem; EKR property; graphs; independent vertex sets 
Academic Unit/School:  Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics Faculty of Science, Technology, Engineering and Mathematics (STEM) 
Item ID:  3524 
Depositing User:  Fred Holroyd 
Date Deposited:  27 Jun 2006 
Last Modified:  19 Feb 2018 02:40 
URI:  http://oro.open.ac.uk/id/eprint/3524 
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