Perfect matchings in planar cubic graphs

Abstract

A well-known conjecture of Lovász and Plummer from the mid-1970’s, still open, asserts that for every cubic graph G with no cutedge, the number of perfect matchings in G is exponential in |V (G)|. In this paper we prove the conjecture for planar graphs; we prove that if G is a planar cubic graph with no cutedge, then G has at least

$$2^{{{\left| {V(G)} \right|} \mathord{\left/ {\vphantom {{\left| {V(G)} \right|} {655978752}}} \right. \kern-\nulldelimiterspace} {655978752}}}$$

perfect matchings.

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Correspondence to Maria Chudnovsky.

Additional information

This research was conducted while the author served as a Clay Mathematics Institute Research Fellow.

Supported by ONR grant N00014-01-1-0608 and NSF grant DMS-0070912.

Note added in proof: The full conjecture by Lovász and Plummer has now been solved, in the following paper: L. Esperet, F. Kardos, A. King, D. Král, and S. Norine, “Exponentially many perfect matchings in cubic graphs”, Advances in Mathematics 227 (2011), 1646–1664.

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Chudnovsky, M., Seymour, P. Perfect matchings in planar cubic graphs. Combinatorica 32, 403–424 (2012). https://doi.org/10.1007/s00493-012-2660-9

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Mathematics Subject Classification (2000)

  • 05C70
  • 05C10