Perfect matchings in planar 3-connected graphs: girth five and minimum degree five

@misc{dai2026perfect,
  title = {{Perfect matchings in planar 3-connected graphs: girth five and minimum degree five}},
  author = {Xinyu Dai},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16648},
  eprint = {2610.16648},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16648},
}

TY  - UNPB
TI  - Perfect matchings in planar 3-connected graphs: girth five and minimum degree five
AU  - Xinyu Dai
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16648
UR  - https://arxiv.org/abs/2610.16648
AB  - Let Φ(G) denote the number of perfect matchings of a graph G. Goedgebeur et al. posted two questions about the perfect matchings of planar 3-connected graphs: does there exist an infinite family of matchable planar 3-connected graphs of girth 5 with a constant number of perfect matchings? Whether there exist infinitely many planar 3-connected graphs with minimum degree 5 which have a constant number of perfect matchings. We present answers to those questions. First, we construct, for every even n≥196, a simple planar 3-connected graph of order n and girth five with exactly c perfect matchings, where c is a fixed positive integer. Second, for a simple planar 3-connected graph G with a perfect matching and minimum degree at least five, we show that Φ(G)≥2^⌈(|V(G)|+20)/22⌉. This implies that, for each positive integer t, only finitely many such graphs, up to isomorphism, have at most t perfect matchings.
ER  -