Construction and characterization of graphs whose each spanning tree has a perfect matching

TL;DR

This paper characterizes all connected graphs where every spanning tree has a perfect matching, using Tutte's theorem and graph structure analysis, and establishes an upper bound on the size of such graphs.

Contribution

It provides a complete characterization of graphs with all spanning trees having perfect matchings and determines the maximum size of such graphs.

Findings

All graphs with each spanning tree having a perfect matching are characterized.

A maximum size bound for these graphs is established as m ≤ (n+1)n/2.

Equality holds if and only if the graph is isomorphic to K_n ◦ K_1.

Abstract

An edge subset $S$ of a connected graph $G$ is called an anti-Kekul\'{e} set if $G-S$ is connected and has no perfect matching. We can see that a connected graph $G$ has no anti-Kekul\'{e} set if and only if each spanning tree of $G$ has a perfect matching. In this paper, by applying Tutte's 1-factor theorem and structure of minimally 2-connected graphs, we characterize all graphs whose each spanning tree has a perfect matching In addition, we show that if $G$ is a connected graph of order $2n$ for a positive integer $n\geq 4$ and size $m$ whose each spanning tree has a perfect matching, then $m\leq \frac{(n+1)n} 2$, with equality if and only if $G\cong K_n\circ K_1$.