Bounding the Number of Edges in Permutation Graphs

Given an integer s ≥ 0 and a permutation π ∈ Sn, let Γπ,s be the graph on n vertices {1, . . . , n} where two vertices i < j are adjacent if the permutation flips their order and there are at most s integers k, i < k < j, such that π = [. . . j . . . k . . . i . . .]. In this short paper we determine the maximum number of edges in Γπ,s for all s ≥ 1 and characterize all permutations π which achieve this maximum. This answers an open question of Adin and Roichman, who studied the case s = 0. We also consider another (closely related) permutation graph, defined by Adin and Roichman, and obtain asymptotically tight bounds on the maximum number of edges in it.