Exponentially many perfect matchings in cubic graphs

Abstract

We show that every cubic bridgeless graph G has at least 2^(|V(G)|/3656) perfect matchings. This confirms an old conjecture of Lovasz and Plummer. This version of the paper uses a different definition of a burl from the journal version of the paper and a different proof of Lemma 18 is given. This simplifies the exposition of our arguments throughout the whole paper

Avoiding 5-Circuits in 2-Factors of Cubic Graphs

“…Many important problems and conjectures can be reduced to snarks: the 4-colour theorem, Tutte's 5-flow conjecture, or the cycle double cover conjecture [6,8]. Problems regarding 1-factors (and thus 2-factors) tend to be challenging; a conjecture that there is exponentially many perfect matchings from 1970s has been proven only recently [4].…”

Spanning bipartite quadrangulations of even triangulations

“…More precisely, for a triangulation , there is a bijection between the spanning quadrangulations in and the perfect matchings in . It is known that the number of perfect matchings of a ‐edge‐connected cubic graph is exponential in the order of . Then, because of the existence of such a huge number of perfect matchings, we expect that there exists a perfect matching of that corresponds to a spanning bipartite quadrangulation, and, respectively, a spanning nonbipartite quadrangulation.…”

“…Theorem 9 (Collins and Hutchinson, [7] and Yeh and Zhu, [31]) Every 6-regular toroidal triangulation is 4-colorable, with the following exceptions; (3,17), (3,18), (3,25), (4,17), (6,17), (6,25), (6,33), (7,19), (7,25), (7,26), (9,25), (10,25), (10,26), (10, 37), (14, 33)}.…”

Partitioning Surfaces Into Quadrilateral Patches: A Survey

“…This assumption holds in many cases: every triangle mesh without boundary has a dual perfect matching [EKK*11]. For meshes with boundary, however, a perfect matching does not always exist (in particular for meshes with an odd number of triangles).…”