New bounds on the number of Edges of the maximal Permutation Graphs

TL;DR

This paper establishes new bounds on the maximum number of edges in permutation graphs with a given number of vertices, using a novel partitioning method for edge labelings.

Contribution

It introduces a new approach to compute bounds on edges of maximal permutation graphs by partitioning edge labelings and analyzing each part separately.

Findings

Explicit lower bounds derived from partitioning method

Upper bounds established for maximal permutation graphs

Method improves understanding of permutation graph edge limits

Abstract

In this paper, we bound the number of edges of a maximal permutation graph with n vertices. We propose a new method to compute the lower bound by splitting the set of labellings of the edges into six parts, considering one separate problem for each part, explicitly determining the cardinality of four parts and summing up the corresponding values. We finish with an upper bound of the number of edges of a maximal permutation graph.