On the ultimate categorical independence ratio

نویسنده

Ágnes Tóth

چکیده

Brown, Nowakowski and Rall defined the ultimate categorical independence ratio of a graph G as A(G) = lim k→∞ i(G×k), where i(G) = α(G) |V (G)| denotes the independence ratio of a graphG, andG ×k is the kth categorical power of G. Let a(G) = max{ |U | |U |+|NG(U)| : U is an independent set of G}, where NG(U) is the neighborhood of U in G. In this paper we answer a question of Alon and Lubetzky, namely we prove that A(G) = a(G) if a(G) ≤ 1 2 , and A(G) = 1 otherwise. We also discuss some other open problems related to A(G) which are immediately settled by this result.

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