Counting graph orientations with no directed triangles

05/27/2020

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by   Pedro Araújo, et al.

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Alon and Yuster proved that the number of orientations of any n-vertex graph in which every K_3 is transitively oriented is at most 2^⌊ n^2/4⌋ for n ≥ 10^4 and conjectured that the precise lower bound on n should be n ≥ 8. We confirm their conjecture and, additionally, characterize the extremal families by showing that the balanced complete bipartite graph with n vertices is the only n-vertex graph for which there are exactly 2^⌊ n^2/4⌋ such orientations.