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Two Results on the Binding Numbers of Product Graphs

The binding number of a graph G is defined to be the minimum of |N(S)|/|S| taken over all nonempty S ⊆ V (G) such that N(S) 6= V (G). In this paper, two general results for the binding numbers of product graphs are obtained. (1) For any G on m vertices, it is shown that bind(G × Kn) = (nm − 1)/(nm − δ(G) − n + 1) for all n sufficiently large. (2) For arbitrary G and for H with bind(H) ≥ 1, a (relatively) simple expression is derived for bind(G[H]). This research formed part of the first author’s Ph.D. thesis and was supported by a grant from the Foundation for Research Development.