Bounding the Number of Edges in Permutation Graphs

Given an integer s 0 and a permutation Sn, let ,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 = [. . . j . . . k . . . i . . .]. In this short paper we determine the maximum number of edges in ,s for all s 1 and characterize all permutations 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.

• Type: Journal
• Year: 2006
• Where: COMBINATORICS
• Authors: Peter Keevash, Po-Shen Loh, Benny Sudakov
• Keywords: COMBINATORICS 2006, Integer, Permutation, Permutation Sn