A note on maximal plane subgraphs of the complete twisted graph containing perfect matchings

TL;DR

This paper investigates the structure of maximal plane subgraphs of a specific complete graph drawing, showing they can be transformed into each other through a sequence of single edge exchanges while maintaining perfect matchings.

Contribution

It establishes that all such maximal plane subgraphs containing perfect matchings are connected via simple edge exchanges, extending understanding of their combinatorial structure.

Findings

Any two maximal plane subgraphs with perfect matchings can be connected through a sequence of single edge exchanges.

The sequence of transformations preserves the property of containing perfect matchings.

The result applies specifically to the twisted graph $T_{n}$ and its maximal plane subgraphs.

Abstract

The twisted graph $T_{n}$ is a drawing of the complete graph with $n$ vertices $v_{1},v_{2},\ldots ,v_{n}$ in which two edges $v_{i}v_{j}$ ($i<j$) and $v_{s}v_{t}$ ($s<t$) cross if and only if $i<s<t<j$ or $s<i<j<t$. We show that for any maximal plane subgraphs $S$ and $R$ of $T_{n}$, each containing at least one perfect matching, there is a sequence $S=F_0, F_1, \ldots, F_m=R$ of maximal plane subgrahs of $T_n$, also containing perfect matchings, such that for $i=0,1, \ldots, m-1$, $F_{i+1}$ can be obtained from $F_{i}$ by a single edge exchange.