58 research outputs found

    A sequence of triangle-free pseudorandom graphs

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    A construction of Alon yields a sequence of highly pseudorandom triangle-free graphs with edge density significantly higher than one might expect from comparison with random graphs. We give an alternative construction for such graphs.Comment: 6 page

    Orthonormal representations of HH-free graphs

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    Let x1,,xnRdx_1, \ldots, x_n \in \mathbb{R}^d be unit vectors such that among any three there is an orthogonal pair. How large can nn be as a function of dd, and how large can the length of x1++xnx_1 + \ldots + x_n be? The answers to these two celebrated questions, asked by Erd\H{o}s and Lov\'{a}sz, are closely related to orthonormal representations of triangle-free graphs, in particular to their Lov\'{a}sz ϑ\vartheta-function and minimum semidefinite rank. In this paper, we study these parameters for general HH-free graphs. In particular, we show that for certain bipartite graphs HH, there is a connection between the Tur\'{a}n number of HH and the maximum of ϑ(G)\vartheta \left( \overline{G} \right) over all HH-free graphs GG.Comment: 16 page

    The Shannon capacity of a graph and the independence numbers of its powers

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    The independence numbers of powers of graphs have been long studied, under several definitions of graph products, and in particular, under the strong graph product. We show that the series of independence numbers in strong powers of a fixed graph can exhibit a complex structure, implying that the Shannon Capacity of a graph cannot be approximated (up to a sub-polynomial factor of the number of vertices) by any arbitrarily large, yet fixed, prefix of the series. This is true even if this prefix shows a significant increase of the independence number at a given power, after which it stabilizes for a while

    Chromatic number, clique subdivisions, and the conjectures of Haj\'os and Erd\H{o}s-Fajtlowicz

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    For a graph GG, let χ(G)\chi(G) denote its chromatic number and σ(G)\sigma(G) denote the order of the largest clique subdivision in GG. Let H(n) be the maximum of χ(G)/σ(G)\chi(G)/\sigma(G) over all nn-vertex graphs GG. A famous conjecture of Haj\'os from 1961 states that σ(G)χ(G)\sigma(G) \geq \chi(G) for every graph GG. That is, H(n)1H(n) \leq 1 for all positive integers nn. This conjecture was disproved by Catlin in 1979. Erd\H{o}s and Fajtlowicz further showed by considering a random graph that H(n)cn1/2/lognH(n) \geq cn^{1/2}/\log n for some absolute constant c>0c>0. In 1981 they conjectured that this bound is tight up to a constant factor in that there is some absolute constant CC such that χ(G)/σ(G)Cn1/2/logn\chi(G)/\sigma(G) \leq Cn^{1/2}/\log n for all nn-vertex graphs GG. In this paper we prove the Erd\H{o}s-Fajtlowicz conjecture. The main ingredient in our proof, which might be of independent interest, is an estimate on the order of the largest clique subdivision which one can find in every graph on nn vertices with independence number α\alpha.Comment: 14 page

    A construction for clique-free pseudorandom graphs

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    A construction of Alon and Krivelevich gives highly pseudorandom Kk-free graphs on n vertices with edge density equal to Θ(n−1=(k−2)). In this short note we improve their result by constructing an infinite family of highly pseudorandom Kk-free graphs with a higher edge density of Θ(n−1=(k−1))
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