7 research outputs found

    A Tutte polynomial inequality for lattice path matroids

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    Let MM be a matroid without loops or coloops and let T(M;x,y)T(M;x,y) be its Tutte polynomial. In 1999 Merino and Welsh conjectured that max⁥(T(M;2,0),T(M;0,2))≄T(M;1,1)\max(T(M;2,0), T(M;0,2))\geq T(M;1,1) holds for graphic matroids. Ten years later, Conde and Merino proposed a multiplicative version of the conjecture which implies the original one. In this paper we prove the multiplicative conjecture for the family of lattice path matroids (generalizing earlier results on uniform and Catalan matroids). In order to do this, we introduce and study particular lattice path matroids, called snakes, used as building bricks to indeed establish a strengthening of the multiplicative conjecture as well as a complete characterization of the cases in which equality holds.Comment: 17 pages, 9 figures, improved exposition/minor correction

    On the number of spanning trees in random regular graphs

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    Let d≄3d \geq 3 be a fixed integer. We give an asympotic formula for the expected number of spanning trees in a uniformly random dd-regular graph with nn vertices. (The asymptotics are as n→∞n\to\infty, restricted to even nn if dd is odd.) We also obtain the asymptotic distribution of the number of spanning trees in a uniformly random cubic graph, and conjecture that the corresponding result holds for arbitrary (fixed) dd. Numerical evidence is presented which supports our conjecture.Comment: 26 pages, 1 figure. To appear in the Electronic Journal of Combinatorics. This version addresses referee's comment

    The Merino–Welsh conjecture holds for series–parallel graphs

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    The Merino–Welsh conjecture asserts that the number of spanning trees of a graph is no greater than the maximum of the numbers of totally cyclic orientations and acyclic orientations of that graph. We prove this conjecture for the class of series–parallel graphs
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