52 research outputs found

    Extensions of Extremal Graph Theory to Grids

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    We consider extensions of Turán\u27s original theorem of 1941 to planar grids. For a complete kxm array of vertices, we establish in Proposition 4.3 an exact formula for the maximal number of edges possible without any square regions. We establish with Theorem 4.12 an upper bound and with Theorem 4.15 an asymptotic lower bound for the maximal number of edges on a general grid graph with n vertices and no rectangles

    On Some Applications of Graph Theory, I

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    In a series of papers, of which the present one is Part 1, it is shown that solutions to a variety of problems in distance geometry, potential theory and theory of metric spaces are provided by appropriate applications of graph theoretic results. (c) 1972 Published by Elsevier B.V

    Some Ramsey- and anti-Ramsey-type results in combinatorial number theory and geometry

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    A szerző nem járult hozzá nyilatkozatában a dolgozat nyilvánosságra hozásához

    On Minrank and Forbidden Subgraphs

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    The minrank over a field F\mathbb{F} of a graph GG on the vertex set {1,2,,n}\{1,2,\ldots,n\} is the minimum possible rank of a matrix MFn×nM \in \mathbb{F}^{n \times n} such that Mi,i0M_{i,i} \neq 0 for every ii, and Mi,j=0M_{i,j}=0 for every distinct non-adjacent vertices ii and jj in GG. For an integer nn, a graph HH, and a field F\mathbb{F}, let g(n,H,F)g(n,H,\mathbb{F}) denote the maximum possible minrank over F\mathbb{F} of an nn-vertex graph whose complement contains no copy of HH. In this paper we study this quantity for various graphs HH and fields F\mathbb{F}. For finite fields, we prove by a probabilistic argument a general lower bound on g(n,H,F)g(n,H,\mathbb{F}), which yields a nearly tight bound of Ω(n/logn)\Omega(\sqrt{n}/\log n) for the triangle H=K3H=K_3. For the real field, we prove by an explicit construction that for every non-bipartite graph HH, g(n,H,R)nδg(n,H,\mathbb{R}) \geq n^\delta for some δ=δ(H)>0\delta = \delta(H)>0. As a by-product of this construction, we disprove a conjecture of Codenotti, Pudl\'ak, and Resta. The results are motivated by questions in information theory, circuit complexity, and geometry.Comment: 15 page

    On the Ramsey-Tur\'an density of triangles

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    One of the oldest results in modern graph theory, due to Mantel, asserts that every triangle-free graphs on nn vertices has at most n2/4\lfloor n^2/4\rfloor edges. About half a century later Andr\'asfai studied dense triangle-free graphs and proved that the largest triangle-free graphs on nn vertices without independent sets of size αn\alpha n, where 2/5α<1/22/5\le \alpha < 1/2, are blow-ups of the pentagon. More than 50 further years have elapsed since Andr\'asfai's work. In this article we make the next step towards understanding the structure of dense triangle-free graphs without large independent sets. Notably, we determine the maximum size of triangle-free graphs~GG on nn vertices with α(G)3n/8\alpha (G)\ge 3n/8 and state a conjecture on the structure of the densest triangle-free graphs GG with α(G)>n/3\alpha(G) > n/3. We remark that the case α(G)n/3\alpha(G) \le n/3 behaves differently, but due to the work of Brandt this situation is fairly well understood.Comment: Revised according to referee report

    On Grids in Point-Line Arrangements in the Plane

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    The famous Szemer\'{e}di-Trotter theorem states that any arrangement of nn points and nn lines in the plane determines O(n4/3)O(n^{4/3}) incidences, and this bound is tight. In this paper, we prove the following Tur\'an-type result for point-line incidence. Let L1\mathcal{L}_1 and L2\mathcal{L}_2 be two sets of tt lines in the plane and let P={12:1L1,2L2}P=\{\ell_1 \cap \ell_2 : \ell_1 \in \mathcal{L}_1, \ell_2 \in \mathcal{L}_2\} be the set of intersection points between L1\mathcal{L}_1 and L2\mathcal{L}_2. We say that (P,L1L2)(P, \mathcal{L}_1 \cup \mathcal{L}_2) forms a \emph{natural t×tt\times t grid} if P=t2|P| =t^2, and conv(P)conv(P) does not contain the intersection point of some two lines in Li,\mathcal{L}_i, for i=1,2.i = 1,2. For fixed t>1t > 1, we show that any arrangement of nn points and nn lines in the plane that does not contain a natural t×tt\times t grid determines O(n43ε)O(n^{\frac{4}{3}- \varepsilon}) incidences, where ε=ε(t)\varepsilon = \varepsilon(t). We also provide a construction of nn points and nn lines in the plane that does not contain a natural 2×22 \times 2 grid and determines at least Ω(n1+114)\Omega({n^{1+\frac{1}{14}}}) incidences.Comment: 13 pages, 5 figure

    On triangle-free graphs maximizing embeddings of bipartite graphs

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    In 1991 Gy\H ori, Pach, and Simonovits proved that for any bipartite graph HH containing a matching avoiding at most 1 vertex, the maximum number of copies of HH in any large enough triangle-free graph is achieved in a balanced complete bipartite graph. In this paper we improve their result by showing that if HH is a bipartite graph containing a matching of size xx and at most 12x1\frac{1}{2}\sqrt{x-1} unmatched vertices, then the maximum number of copies of HH in any large enough triangle-free graph is achieved in a complete bipartite graph. We also prove that such a statement cannot hold if the number of unmatched vertices is Ω(x)\Omega(x)

    On some applications of graph theory, I.

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    In a series of papers, of which the present one is Part I, it is shown that solutions to a variety of problems in distance geometry, potential theory and theory of metric spaces are provided by appropriate applications of graph theoretic results. © 1972
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