The number of perfect matchings in 3-connected planar graphs

Wuxian Chen; Xinyu Dai; Fuliang Lu

The number of perfect matchings in 3-connected planar graphs

Combinatorics 2026-07-19 v1

Authors: Wuxian Chen , Xinyu Dai , Fuliang Lu

Abstract

A graph is matchable if it admits a perfect matching. Recently, Goedgebeur et al. asked whether there exists a constant $c<12$ such that infinitely many matchable planar $3$ -connected graphs, each with exactly $c$ perfect matchings. We prove that every matchable planar $3$ -connected graphs on at least 40 vertices has at least 12 perfect matchings, and this lower bound is sharp. This answers the question negatively.

Keywords

perfect matchings planar graphs 3-connected graphs matchable graphs lower bound

Cite

@article{arxiv.2607.17309,
  title  = {The number of perfect matchings in 3-connected planar graphs},
  author = {Wuxian Chen and Xinyu Dai and Fuliang Lu},
  journal= {arXiv preprint arXiv:2607.17309},
  year   = {2026}
}

Comments

12 pages,5 figures