Perfect matchings of line graphs with small maximum degree

TL;DR

This paper derives a formula for counting perfect matchings in line graphs of certain graphs with degrees 2 or 3, confirming a conjecture for cubic line graphs and applying results to lattice structures in physics.

Contribution

It provides a new explicit formula for perfect matchings in line graphs with small maximum degree, confirming a conjecture for connected cubic line graphs.

Findings

Number of perfect matchings in line graphs with even edges is $2^{n/2+1}$.

Confirmed Lovász and Plummer's conjecture for connected cubic line graphs.

Enumerated perfect matchings in Kagomé, $3.12.12$, and Sierpinski lattices.

Abstract

Let $G$ be a connected graph with vertex set $V(G)=\{v_1,v_2,...,v_{\nu}\}$, which may have multiple edges but have no loops, and $2\leq d_G(v_i)\leq 3$ for $i=1,2,...,\nu$, where $d_G(v)$ denotes the degree of vertex $v$ of $G$. We show that if $G$ has an even number of edges, then the number of perfect matchings of the line graph of $G$ equals $2^{n/2+1}$, where $n$ is the number of 3-degree vertices of $G$. As a corollary, we prove that the number of perfect matchings of a connected cubic line graph with $n$ vertices equals $2^{n/6+1}$ if $n>4$, which implies the conjecture by Lov\'asz and Plummer holds for the connected cubic line graphs. As applications, we enumerate perfect matchings of the Kagom\'e lattices, $3.12.12$ lattices, and Sierpinski gasket with dimension two in the context of statistical physics.