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Independent sets in tensor graph powers. (English) Zbl 1108.05068
Authors’ abstract: The tensor product of two graphs, $$G$$ and $$H$$, has a vertex set $$V(G) \times V(H)$$ and an edge between $$(u,v)$$ and $$(u',v')$$ if and only if both $$u\, u'\in E(G)$$ and $$v\, v'\in E(H)$$. Let $$A(G)$$ denote the limit of the independence ratios of tensor powers of $$G, \lim \alpha (G^n)/|V(G^n)|$$. This parameter was introduced in [J. I. Brown, R. J. Nowakowski and D. Rall, SIAM J. Discrete Math. 9, No. 2, 290–300 (1996; Zbl 0848.05036)], where it was shown that $$A(G)$$ is lower bounded by the vertex expansion ratio of independent sets of $$G$$. In this article we study the relation between these parameters further, and ask whether they are in fact equal. We present several families of graphs where equality holds, and discuss the effect the above question has on various open problems related to tensor graph products.

MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
Keywords:
tensor product; independence ratio
Zbl 0848.05036
Full Text:
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