63 research outputs found

    Distinguishing graphs by their left and right homomorphism profiles

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    We introduce a new property of graphs called ‘q-state Potts unique-ness’ and relate it to chromatic and Tutte uniqueness, and also to ‘chromatic–flow uniqueness’, recently studied by Duan, Wu and Yu. We establish for which edge-weighted graphs H homomor-phism functions from multigraphs G to H are specializations of the Tutte polynomial of G, in particular answering a question of Freed-man, Lovász and Schrijver. We also determine for which edge-weighted graphs H homomorphism functions from multigraphs G to H are specializations of the ‘edge elimination polynomial’ of Averbouch, Godlin and Makowsky and the ‘induced subgraph poly-nomial’ of Tittmann, Averbouch and Makowsky. Unifying the study of these and related problems is the notion of the left and right homomorphism profiles of a graph.Ministerio de Educación y Ciencia MTM2008-05866-C03-01Junta de Andalucía FQM- 0164Junta de Andalucía P06-FQM-0164

    The equivariant topology of stable Kneser graphs

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    The stable Kneser graph SGn,kSG_{n,k}, n1n\ge1, k0k\ge0, introduced by Schrijver \cite{schrijver}, is a vertex critical graph with chromatic number k+2k+2, its vertices are certain subsets of a set of cardinality m=2n+km=2n+k. Bj\"orner and de Longueville \cite{anders-mark} have shown that its box complex is homotopy equivalent to a sphere, \Hom(K_2,SG_{n,k})\homot\Sphere^k. The dihedral group D2mD_{2m} acts canonically on SGn,kSG_{n,k}, the group C2C_2 with 2 elements acts on K2K_2. We almost determine the (C2×D2m)(C_2\times D_{2m})-homotopy type of \Hom(K_2,SG_{n,k}) and use this to prove the following results. The graphs SG2s,4SG_{2s,4} are homotopy test graphs, i.e. for every graph HH and r0r\ge0 such that \Hom(SG_{2s,4},H) is (r1)(r-1)-connected, the chromatic number χ(H)\chi(H) is at least r+6r+6. If k{0,1,2,4,8}k\notin\set{0,1,2,4,8} and nN(k)n\ge N(k) then SGn,kSG_{n,k} is not a homotopy test graph, i.e.\ there are a graph GG and an r1r\ge1 such that \Hom(SG_{n,k}, G) is (r1)(r-1)-connected and χ(G)<r+k+2\chi(G)<r+k+2.Comment: 34 pp

    Logical Equivalences, Homomorphism Indistinguishability, and Forbidden Minors

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    Two graphs GG and HH are homomorphism indistinguishable over a class of graphs F\mathcal{F} if for all graphs FFF \in \mathcal{F} the number of homomorphisms from FF to GG is equal to the number of homomorphisms from FF to HH. Many natural equivalence relations comparing graphs such as (quantum) isomorphism, spectral, and logical equivalences can be characterised as homomorphism indistinguishability relations over certain graph classes. Abstracting from the wealth of such instances, we show in this paper that equivalences w.r.t. any self-complementarity logic admitting a characterisation as homomorphism indistinguishability relation can be characterised by homomorphism indistinguishability over a minor-closed graph class. Self-complementarity is a mild property satisfied by most well-studied logics. This result follows from a correspondence between closure properties of a graph class and preservation properties of its homomorphism indistinguishability relation. Furthermore, we classify all graph classes which are in a sense finite (essentially profinite) and satisfy the maximality condition of being homomorphism distinguishing closed, i.e. adding any graph to the class strictly refines its homomorphism indistinguishability relation. Thereby, we answer various question raised by Roberson (2022) on general properties of the homomorphism distinguishing closure.Comment: 26 pages, 1 figure, 1 tabl

    Measures on the square as sparse graph limits

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    We study a metric on the set of finite graphs in which two graphs are considered to be similar if they have similar bounded dimensional "factors". We show that limits of convergent graph sequences in this metric can be represented by symmetric Borel measures on [0, 1](2). This leads to a generalization of dense graph limit theory to sparse graph sequences. (C) 2019 Elsevier Inc. All rights reserved

    Upper tails and independence polynomials in random graphs

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    The upper tail problem in the Erd\H{o}s--R\'enyi random graph GGn,pG\sim\mathcal{G}_{n,p} asks to estimate the probability that the number of copies of a graph HH in GG exceeds its expectation by a factor 1+δ1+\delta. Chatterjee and Dembo showed that in the sparse regime of p0p\to 0 as nn\to\infty with pnαp \geq n^{-\alpha} for an explicit α=αH>0\alpha=\alpha_H>0, this problem reduces to a natural variational problem on weighted graphs, which was thereafter asymptotically solved by two of the authors in the case where HH is a clique. Here we extend the latter work to any fixed graph HH and determine a function cH(δ)c_H(\delta) such that, for pp as above and any fixed δ>0\delta>0, the upper tail probability is exp[(cH(δ)+o(1))n2pΔlog(1/p)]\exp[-(c_H(\delta)+o(1))n^2 p^\Delta \log(1/p)], where Δ\Delta is the maximum degree of HH. As it turns out, the leading order constant in the large deviation rate function, cH(δ)c_H(\delta), is governed by the independence polynomial of HH, defined as PH(x)=iH(k)xkP_H(x)=\sum i_H(k) x^k where iH(k)i_H(k) is the number of independent sets of size kk in HH. For instance, if HH is a regular graph on mm vertices, then cH(δ)c_H(\delta) is the minimum between 12δ2/m\frac12 \delta^{2/m} and the unique positive solution of PH(x)=1+δP_H(x) = 1+\delta
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