2,815 research outputs found

    A note on coloring vertex-transitive graphs

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    We prove bounds on the chromatic number χ\chi of a vertex-transitive graph in terms of its clique number ω\omega and maximum degree Δ\Delta. We conjecture that every vertex-transitive graph satisfies χmax{ω,5Δ+36}\chi \le \max \left\{\omega, \left\lceil\frac{5\Delta + 3}{6}\right\rceil\right\} and we prove results supporting this conjecture. Finally, for vertex-transitive graphs with Δ13\Delta \ge 13 we prove the Borodin-Kostochka conjecture, i.e., χmax{ω,Δ1}\chi\le\max\{\omega,\Delta-1\}

    Ramsey-nice families of graphs

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    For a finite family F\mathcal{F} of fixed graphs let Rk(F)R_k(\mathcal{F}) be the smallest integer nn for which every kk-coloring of the edges of the complete graph KnK_n yields a monochromatic copy of some FFF\in\mathcal{F}. We say that F\mathcal{F} is kk-nice if for every graph GG with χ(G)=Rk(F)\chi(G)=R_k(\mathcal{F}) and for every kk-coloring of E(G)E(G) there exists a monochromatic copy of some FFF\in\mathcal{F}. It is easy to see that if F\mathcal{F} contains no forest, then it is not kk-nice for any kk. It seems plausible to conjecture that a (weak) converse holds, namely, for any finite family of graphs F\mathcal{F} that contains at least one forest, and for all kk0(F)k\geq k_0(\mathcal{F}) (or at least for infinitely many values of kk), F\mathcal{F} is kk-nice. We prove several (modest) results in support of this conjecture, showing, in particular, that it holds for each of the three families consisting of two connected graphs with 3 edges each and observing that it holds for any family F\mathcal{F} containing a forest with at most 2 edges. We also study some related problems and disprove a conjecture by Aharoni, Charbit and Howard regarding the size of matchings in regular 3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure

    Every planar graph with the Liouville property is amenable

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    We introduce a strengthening of the notion of transience for planar maps in order to relax the standard condition of bounded degree appearing in various results, in particular, the existence of Dirichlet harmonic functions proved by Benjamini and Schramm. As a corollary we obtain that every planar non-amenable graph admits Dirichlet harmonic functions
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