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    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
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