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
perfect matchings.
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References
T. Fowler: Unique Coloring of Planar Graphs, Ph.D. thesis, Georgia Institute of Technology, 1998.
D. Král, J.-S. Sereni and M. Stiebitz: A new lower bound on the number of perfect matchings in cubic graphs, submitted for publication (manuscript 2008).
L. Lovász and M. Plummer: Matching Theory, Annals of Discrete Math. 29, North Holland, Amsterdam, 1986.
A. Schrijver: Counting 1-factors in regular bipartite graphs, J. Combinatorial Theory, Ser. B 72 (1998), 122–135.
H. Whitney: A theorem on graphs, Ann. Math. 32 (1931), 378–390.
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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.