18,290 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 γ∼1.14196\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

    Combinatorics and Geometry of Transportation Polytopes: An Update

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    A transportation polytope consists of all multidimensional arrays or tables of non-negative real numbers that satisfy certain sum conditions on subsets of the entries. They arise naturally in optimization and statistics, and also have interest for discrete mathematics because permutation matrices, latin squares, and magic squares appear naturally as lattice points of these polytopes. In this paper we survey advances on the understanding of the combinatorics and geometry of these polyhedra and include some recent unpublished results on the diameter of graphs of these polytopes. In particular, this is a thirty-year update on the status of a list of open questions last visited in the 1984 book by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure

    Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs

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    The chromatic polynomial of a graph G counts the number of proper colorings of G. We give an affirmative answer to the conjecture of Read and Rota-Heron-Welsh that the absolute values of the coefficients of the chromatic polynomial form a log-concave sequence. We define a sequence of numerical invariants of projective hypersurfaces analogous to the Milnor number of local analytic hypersurfaces. Then we give a characterization of correspondences between projective spaces up to a positive integer multiple which includes the conjecture on the chromatic polynomial as a special case. As a byproduct of our approach, we obtain an analogue of Kouchnirenko's theorem relating the Milnor number with the Newton polytope.Comment: Improved readability. Final version, to appear in J. Amer. Math. So

    Implementing Brouwer's database of strongly regular graphs

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    Andries Brouwer maintains a public database of existence results for strongly regular graphs on n≤1300n\leq 1300 vertices. We implemented most of the infinite families of graphs listed there in the open-source software Sagemath, as well as provided constructions of the "sporadic" cases, to obtain a graph for each set of parameters with known examples. Besides providing a convenient way to verify these existence results from the actual graphs, it also extends the database to higher values of nn.Comment: 18 pages, LaTe

    Some local--global phenomena in locally finite graphs

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    In this paper we present some results for a connected infinite graph GG with finite degrees where the properties of balls of small radii guarantee the existence of some Hamiltonian and connectivity properties of GG. (For a vertex ww of a graph GG the ball of radius rr centered at ww is the subgraph of GG induced by the set Mr(w)M_r(w) of vertices whose distance from ww does not exceed rr). In particular, we prove that if every ball of radius 2 in GG is 2-connected and GG satisfies the condition dG(u)+dG(v)≥∣M2(w)∣−1d_G(u)+d_G(v)\geq |M_2(w)|-1 for each path uwvuwv in GG, where uu and vv are non-adjacent vertices, then GG has a Hamiltonian curve, introduced by K\"undgen, Li and Thomassen (2017). Furthermore, we prove that if every ball of radius 1 in GG satisfies Ore's condition (1960) then all balls of any radius in GG are Hamiltonian.Comment: 18 pages, 6 figures; journal accepted versio
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