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Register a new userTowards characterizing locally common graphs
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A graph H is common if the number of monochromatic copies of H in a 2-edge-coloring of the complete graph is asymptotically minimized by the random coloring. The classification of common graphs is one of the most intriguing problems in extremal graph theory. We study this notion in the local setting as considered by Csoka, Hubai and Lovasz [arXiv:1912.02926], where the graph is required to be the minimizer with respect to perturbations of the random 2-edge-coloring, and give a complete characterization of graphs H into three categories in regard to a possible behavior of the 12 initial terms in the Taylor series determining the number of monochromatic copies of H in such perturbations: graphs of Class I are locally common, graphs of Class II are not locally common, and graphs of Class III cannot be determined to be locally common or not based on the initial 12 terms. As a corollary, we obtain new necessary conditions on a graph to be common and new sufficient conditions on a graph to be not common.
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A graph is locally irregular if any pair of adjacent vertices have distinct degrees. A locally irregular decomposition of a graph $G$ is a decomposition $mathcal{D}$ of $G$ such that every subgraph $H in mathcal{D}$ is locally irregular. A graph is said to be decomposable if it admits a locally irregular decomposition. We prove that any decomposable split graph can be decomposed into at most three locally irregular subgraphs and we characterize all split graphs whose decomposition can be into one, two or three locally irregular subgraphs.
A graph H is common if the number of monochromatic copies of H in a 2-edge-colouring of the complete graph is minimised by the random colouring. Burr and Rosta, extending a famous conjecture by Erdos, conjectured that every graph is common. The conjectures by Erdos and by Burr and Rosta were disproved by Thomason and by Sidorenko, respectively, in the late 1980s. Collecting new examples for common graphs had not seen much progress since then, although very recently, a few more graphs are verified to be common by the flag algebra method or the recent progress on Sidorenkos conjecture. Our contribution here is to give a new class of tripartite common graphs. The first example class is so-called triangle-trees, which generalises two theorems by Sidorenko and answers a question by Jagger, v{S}v{t}oviv{c}ek, and Thomason from 1996. We also prove that, somewhat surprisingly, given any tree T, there exists a triangle-tree such that the graph obtained by adding T as a pendant tree is still common. Furthermore, we show that adding arbitrarily many apex vertices to any connected bipartite graph on at most five vertices give a common graph.
A graph H is k-common if the number of monochromatic copies of H in a k-edge-coloring of K_n is asymptotically minimized by a random coloring. For every k, we construct a connected non-bipartite k-common graph. This resolves a problem raised by Jagger, Stovicek and Thomason [Combinatorica 16 (1996), 123-141]. We also show that a graph H is k-common for every k if and only if H is Sidorenko and that H is locally k-common for every k if and only if H is locally Sidorenko.
We provide a gentle introduction, aimed at non-experts, to Borel combinatorics that studies definable graphs on topological spaces. This is an emerging field on the borderline between combinatorics and descriptive set theory with deep connections to many other areas. After giving some background material, we present in careful detail some basic tools and results on the existence of Borel satisfying assignments: Bore
A {bf map} is a graph that admits an orientation of its edges so that each vertex has out-degree exactly 1. We characterize graphs which admit a decomposition into $k$ edge-disjoint maps after: (1) the addition of {it any} $ell$ edges; (2) the addition of {it some} $ell$ edges. These graphs are identified with classes of {it sparse} graphs; the results are also given in matroidal terms.
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