Multiple Petersen subdivisions in permutation graphs

TL;DR

This paper explores the structure of permutation graphs, demonstrating that certain edges are contained in Petersen subdivisions, and establishes bounds on the number of such subdivisions in graphs without 4-cycles.

Contribution

It extends previous results by characterizing edges in permutation graphs related to Petersen subdivisions and provides bounds on their quantity in specific graph classes.

Findings

Edges in certain permutation graphs are contained in Petersen subdivisions.

A linear lower bound on Petersen subdivisions in graphs with no 4-cycles.

Construction showing the lower bound is tight up to a constant factor.

Abstract

A permutation graph is a cubic graph admitting a 1-factor M whose complement consists of two chordless cycles. Extending results of Ellingham and of Goldwasser and Zhang, we prove that if e is an edge of M such that every 4-cycle containing an edge of M contains e, then e is contained in a subdivision of the Petersen graph of a special type. In particular, if the graph is cyclically 5-edge-connected, then every edge of M is contained in such a subdivision. Our proof is based on a characterization of cographs in terms of twin vertices. We infer a linear lower bound on the number of Petersen subdivisions in a permutation graph with no 4-cycles, and give a construction showing that this lower bound is tight up to a constant factor.