Perfect Matchings in Claw-free Cubic Graphs

TL;DR

This paper proves that claw-free cubic graphs without cutedges have exponentially many perfect matchings, confirming a special case of a broader conjecture about such matchings in regular graphs.

Contribution

It establishes the first exponential lower bound on the number of perfect matchings in claw-free cubic graphs, verifying Lovasz and Plummer's conjecture for this class.

Findings

Claw-free cubic graphs have more than 2^(n/12) perfect matchings.

The result confirms the conjecture for a new class of graphs.

Provides a significant step towards the general conjecture for all cubic graphs.

Abstract

Lovasz and Plummer conjectured that there exists a fixed positive constant c such that every cubic n-vertex graph with no cutedge has at least 2^(cn) perfect matchings. Their conjecture has been verified for bipartite graphs by Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every claw-free cubic n-vertex graph with no cutedge has more than 2^(n/12) perfect matchings, thus verifying the conjecture for claw-free graphs.