Holroyd, Fred and Talbot, John
Graphs with the Erdos-Ko-Rado property.
Discrete Mathematics, 293(1-3) pp. 165–176.
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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.
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