Counting orientations of graphs with no strongly connected tournaments

01/29/2021

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by   Fábio Botler, et al.

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Let S_k(n) be the maximum number of orientations of an n-vertex graph G in which no copy of K_k is strongly connected. For all integers n, k≥ 4 where n≥ 5 or k≥ 5, we prove that S_k(n) = 2^t_k-1(n), where t_k-1(n) is the number of edges of the n-vertex (k-1)-partite Turán graph T_k-1(n), and that T_k-1(n) is the only n-vertex graph with this number of orientations. Furthermore, S_4(4) = 40 and this maximality is achieved only by K_4.