Abstract:We investigate the minimum non-zero number of perfect matchings in planar graphs. We prove that this is a constant for 2-connected planar graphs of minimum degree 3 and 3-connected planar graphs. In the former case, the constant is 4 and this is best possible. In the 3-connected case, we give several infinite families, including nearly 3-regular graphs and triangulations with a constant number of perfect matchings. In contrast, it was known that in the 4-connected case the minimum non-zero number of perfect matchings is at least linear. For 5-connected triangulations, it follows from a result of Alahmadi, Aldred, and Thomassen that there must be exponentially many perfect matchings. The families of planar graphs we investigate here are classified by connectivity. We conclude the article with an infinite family of counterexamples to a conjecture published by Zaks; these are not planar, but very much concern connectivity constraints.
From: Tibo Van den Eede [view email]
[v1]
Sat, 20 Jun 2026 22:35:00 UTC (22 KB)
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