1,643 research outputs found

    Sidorenko's conjecture, colorings and independent sets

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    Let hom(H,G)\hom(H,G) denote the number of homomorphisms from a graph HH to a graph GG. Sidorenko's conjecture asserts that for any bipartite graph HH, and a graph GG we have hom(H,G)v(G)v(H)(hom(K2,G)v(G)2)e(H),\hom(H,G)\geq v(G)^{v(H)}\left(\frac{\hom(K_2,G)}{v(G)^2}\right)^{e(H)}, where v(H),v(G)v(H),v(G) and e(H),e(G)e(H),e(G) denote the number of vertices and edges of the graph HH and GG, respectively. In this paper we prove Sidorenko's conjecture for certain special graphs GG: for the complete graph KqK_q on qq vertices, for a K2K_2 with a loop added at one of the end vertices, and for a path on 33 vertices with a loop added at each vertex. These cases correspond to counting colorings, independent sets and Widom-Rowlinson colorings of a graph HH. For instance, for a bipartite graph HH the number of qq-colorings ch(H,q)\textrm{ch}(H,q) satisfies ch(H,q)qv(H)(q1q)e(H).\textrm{ch}(H,q)\geq q^{v(H)}\left(\frac{q-1}{q}\right)^{e(H)}. In fact, we will prove that in the last two cases (independent sets and Widom-Rowlinson colorings) the graph HH does not need to be bipartite. In all cases, we first prove a certain correlation inequality which implies Sidorenko's conjecture in a stronger form.Comment: Two references added and Remark 2.1 is expande

    Acyclic edge-coloring using entropy compression

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    An edge-coloring of a graph G is acyclic if it is a proper edge-coloring of G and every cycle contains at least three colors. We prove that every graph with maximum degree Delta has an acyclic edge-coloring with at most 4 Delta - 4 colors, improving the previous bound of 9.62 (Delta - 1). Our bound results from the analysis of a very simple randomised procedure using the so-called entropy compression method. We show that the expected running time of the procedure is O(mn Delta^2 log Delta), where n and m are the number of vertices and edges of G. Such a randomised procedure running in expected polynomial time was only known to exist in the case where at least 16 Delta colors were available. Our aim here is to make a pedagogic tutorial on how to use these ideas to analyse a broad range of graph coloring problems. As an application, also show that every graph with maximum degree Delta has a star coloring with 2 sqrt(2) Delta^{3/2} + Delta colors.Comment: 13 pages, revised versio

    Critical behavior of colored tensor models in the large N limit

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    Colored tensor models have been recently shown to admit a large N expansion, whose leading order encodes a sum over a class of colored triangulations of the D-sphere. The present paper investigates in details this leading order. We show that the relevant triangulations proliferate like a species of colored trees. The leading order is therefore summable and exhibits a critical behavior, independent of the dimension. A continuum limit is reached by tuning the coupling constant to its critical value while inserting an infinite number of pairs of D-simplices glued together in a specific way. We argue that the dominant triangulations are branched polymers.Comment: 20 page

    Approximate Counting, the Lovasz Local Lemma and Inference in Graphical Models

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    In this paper we introduce a new approach for approximately counting in bounded degree systems with higher-order constraints. Our main result is an algorithm to approximately count the number of solutions to a CNF formula Φ\Phi when the width is logarithmic in the maximum degree. This closes an exponential gap between the known upper and lower bounds. Moreover our algorithm extends straightforwardly to approximate sampling, which shows that under Lov\'asz Local Lemma-like conditions it is not only possible to find a satisfying assignment, it is also possible to generate one approximately uniformly at random from the set of all satisfying assignments. Our approach is a significant departure from earlier techniques in approximate counting, and is based on a framework to bootstrap an oracle for computing marginal probabilities on individual variables. Finally, we give an application of our results to show that it is algorithmically possible to sample from the posterior distribution in an interesting class of graphical models.Comment: 25 pages, 2 figure
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