255 research outputs found

    On the expected number of perfect matchings in cubic planar graphs

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    A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that a bridgeless cubic graph has exponentially many perfect matchings. It was solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the special case of cubic planar graphs. In our work we consider random bridgeless cubic planar graphs with the uniform distribution on graphs with nn vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically cγnc\gamma^n, where c>0c>0 and γ1.14196\gamma \sim 1.14196 is an explicit algebraic number. We also compute the expected number of perfect matchings in (non necessarily bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs. Our starting point is a correspondence between counting perfect matchings in rooted cubic planar maps and the partition function of the Ising model in rooted triangulations.Comment: 19 pages, 4 figure

    On the expected number of perfect matchings in cubic planar graphs

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    A well-known conjecture by Lov'asz and Plummer from the 1970s asserted that a bridgeless cubic graph has exponentially many perfect matchings. It was solved in the affirmative by Esperet et al. ([13]). On the other hand, Chudnovsky and Seymour ([8]) proved the conjecture in the special case of cubic planar graphs. In our work we consider random bridgeless cubic planar graphs with the uniform distribution on graphs with n vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically cγn, where c > 0 and γ ∼ 1.14196 is an explicit algebraic number. We also compute the expected number of perfect matchings in (not necessarily bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs. Our starting point is a correspondence between counting perfect matchings in rooted cubic planar maps and the partition function of the Ising model in rooted triangulations

    Flows and bisections in cubic graphs

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    A kk-weak bisection of a cubic graph GG is a partition of the vertex-set of GG into two parts V1V_1 and V2V_2 of equal size, such that each connected component of the subgraph of GG induced by ViV_i (i=1,2i=1,2) is a tree of at most k2k-2 vertices. This notion can be viewed as a relaxed version of nowhere-zero flows, as it directly follows from old results of Jaeger that every cubic graph GG with a circular nowhere-zero rr-flow has a r\lfloor r \rfloor-weak bisection. In this paper we study problems related to the existence of kk-weak bisections. We believe that every cubic graph which has a perfect matching, other than the Petersen graph, admits a 4-weak bisection and we present a family of cubic graphs with no perfect matching which do not admit such a bisection. The main result of this article is that every cubic graph admits a 5-weak bisection. When restricted to bridgeless graphs, that result would be a consequence of the assertion of the 5-flow Conjecture and as such it can be considered a (very small) step toward proving that assertion. However, the harder part of our proof focuses on graphs which do contain bridges.Comment: 14 pages, 6 figures - revised versio

    A superlinear bound on the number of perfect matchings in cubic bridgeless graphs

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    Lovasz and Plummer conjectured in the 1970's that cubic bridgeless graphs have exponentially many perfect matchings. This conjecture has been verified for bipartite graphs by Voorhoeve in 1979, and for planar graphs by Chudnovsky and Seymour in 2008, but in general only linear bounds are known. In this paper, we provide the first superlinear bound in the general case.Comment: 54 pages v2: a short (missing) proof of Lemma 10 was adde

    Some snarks are worse than others

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    Many conjectures and open problems in graph theory can either be reduced to cubic graphs or are directly stated for cubic graphs. Furthermore, it is known that for a lot of problems, a counterexample must be a snark, i.e. a bridgeless cubic graph which is not 3--edge-colourable. In this paper we deal with the fact that the family of potential counterexamples to many interesting conjectures can be narrowed even further to the family S5{\cal S}_{\geq 5} of bridgeless cubic graphs whose edge set cannot be covered with four perfect matchings. The Cycle Double Cover Conjecture, the Shortest Cycle Cover Conjecture and the Fan-Raspaud Conjecture are examples of statements for which S5{\cal S}_{\geq 5} is crucial. In this paper, we study parameters which have the potential to further refine S5{\cal S}_{\geq 5} and thus enlarge the set of cubic graphs for which the mentioned conjectures can be verified. We show that S5{\cal S}_{\geq 5} can be naturally decomposed into subsets with increasing complexity, thereby producing a natural scale for proving these conjectures. More precisely, we consider the following parameters and questions: given a bridgeless cubic graph, (i) how many perfect matchings need to be added, (ii) how many copies of the same perfect matching need to be added, and (iii) how many 2--factors need to be added so that the resulting regular graph is Class I? We present new results for these parameters and we also establish some strong relations between these problems and some long-standing conjectures.Comment: 27 pages, 16 figure

    On Zero-free Intervals of Flow Polynomials

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    This article studies real roots of the flow polynomial F(G,λ)F(G,\lambda) of a bridgeless graph GG. For any integer k0k\ge 0, let ξk\xi_k be the supremum in (1,2](1,2] such that F(G,λ)F(G,\lambda) has no real roots in (1,ξk)(1,\xi_k) for all graphs GG with W(G)k|W(G)|\le k, where W(G)W(G) is the set of vertices in GG of degrees larger than 33. We prove that ξk\xi_k can be determined by considering a finite set of graphs and show that ξk=2\xi_k=2 for k2k\le 2, ξ3=1.430\xi_3=1.430\cdots, ξ4=1.361\xi_4=1.361\cdots and ξ5=1.317\xi_5=1.317\cdots. We also prove that for any bridgeless graph G=(V,E)G=(V,E), if all roots of F(G,λ)F(G,\lambda) are real but some of these roots are not in the set {1,2,3}\{1,2,3\}, then EV+17|E|\ge |V|+17 and F(G,λ)F(G,\lambda) has at least 9 real roots in (1,2)(1,2).Comment: 26 pages, 7 figure

    The Cost of Perfection for Matchings in Graphs

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    Perfect matchings and maximum weight matchings are two fundamental combinatorial structures. We consider the ratio between the maximum weight of a perfect matching and the maximum weight of a general matching. Motivated by the computer graphics application in triangle meshes, where we seek to convert a triangulation into a quadrangulation by merging pairs of adjacent triangles, we focus mainly on bridgeless cubic graphs. First, we characterize graphs that attain the extreme ratios. Second, we present a lower bound for all bridgeless cubic graphs. Third, we present upper bounds for subclasses of bridgeless cubic graphs, most of which are shown to be tight. Additionally, we present tight bounds for the class of regular bipartite graphs
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