12 research outputs found

    Vertex colouring and forbidden subgraphs - a survey

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    There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions

    On the equivalence covering number of splitgraphs

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    Extremal problems on counting combinatorial structures

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    The fast developing field of extremal combinatorics provides a diverse spectrum of powerful tools with many applications to economics, computer science, and optimization theory. In this thesis, we focus on counting and coloring problems in this field. The complete balanced bipartite graph on nn vertices has \floor{n^2/4} edges. Since all of its subgraphs are triangle-free, the number of (labeled) triangle-free graphs on nn vertices is at least 2^{\floor{n^2/4}}. This was shown to be the correct order of magnitude in a celebrated paper Erd\H{o}s, Kleitman, and Rothschild from 1976, where the authors furthermore proved that almost all triangle-free graphs are bipartite. In Chapters 2 and 3 we study analogous problems for triangle-free graphs that are maximal with respect to inclusion. In Chapter 2, we solve the following problem of Paul Erd\H{o}s: Determine or estimate the number of maximal triangle-free graphs on nn vertices. We show that the number of maximal triangle-free graphs is at most 2n2/8+o(n2)2^{n^2/8+o(n^2)}, which matches the previously known lower bound. Our proof uses among other tools the Ruzsa-Szemer\'{e}di Triangle Removal Lemma and recent results on characterizing of the structure of independent sets in hypergraphs. This is a joint work with J\'{o}zsef Balogh. In Chapter 3, we investigate the structure of maximal triangle-free graphs. We prove that almost all maximal triangle-free graphs admit a vertex partition (X,Y)(X, Y) such that G[X]G[X] is a perfect matching and YY is an independent set. Our proof uses the Ruzsa-Szemer\'{e}di Removal Lemma, the Erd\H{o}s-Simonovits stability theorem, and recent results of Balogh-Morris-Samotij and Saxton-Thomason on the characterization of the structure of independent sets in hypergraphs. The proof also relies on a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint P3P_3's, which is of independent interest. This is a joint work with J\'{o}zsef Balogh, Hong Liu, and Maryam Sharifzadeh. In Chapte 4, we seek families in posets with the smallest number of comparable pairs. Given a poset PP, a family \F\subseteq P is \emph{centered} if it is obtained by `taking sets as close to the middle layer as possible'. A poset PP is said to have the \emph{centeredness property} if for any MM, among all families of size MM in PP, centered families contain the minimum number of comparable pairs. Kleitman showed that the Boolean lattice {0,1}n\{0,1\}^n has the centeredness property. It was conjectured by Noel, Scott, and Sudakov, and by Balogh and Wagner, that the poset {0,1,,k}n\{0,1,\ldots,k\}^n also has the centeredness property, provided nn is sufficiently large compared to kk. We show that this conjecture is false for all k2k\geq 2 and investigate the range of MM for which it holds. Further, we improve a result of Noel, Scott, and Sudakov by showing that the poset of subspaces of Fqn\mathbb{F}_q^n has the centeredness property. Several open problems are also given. This is a joint result with J\'{o}zsef Balogh and Adam Zsolt Wagner. In Chapter 5, we consider a graph coloring problem. Kim and Park have found an infinite family of graphs whose squares are not chromatic-choosable. Xuding Zhu asked whether there is some kk such that all kk-th power graphs are chromatic-choosable. We answer this question in the negative: we show that there is a positive constant cc such that for any kk there is a family of graphs GG with χ(Gk)\chi(G^k) unbounded and χ(Gk)cχ(Gk)logχ(Gk)\chi_{\ell}(G^k)\geq c \chi(G^k) \log \chi(G^k). We also provide an upper bound, χ(Gk)1\chi_{\ell}(G^k)1. This is a joint work with Nicholas Kosar, Benjamin Reiniger, and Elyse Yeager

    Simulated annealing based datapath synthesis

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    Packing and covering in combinatorics

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    Distance measures in graphs and subgraphs.

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    Thesis (M.Sc.)-University of Natal, 1996.In this thesis we investigate how the modification of a graph affects various distance measures. The questions considered arise in the study of how the efficiency of communications networks is affected by the loss of links or nodes. In a graph C, the distance between two vertices is the length of a shortest path between them. The eccentricity of a vertex v is the maximum distance from v to any vertex in C. The radius of C is the minimum eccentricity of a vertex, and the diameter of C is the maximum eccentricity of a vertex. The distance of C is defined as the sum of the distances between all unordered pairs of vertices. We investigate, for each of the parameters radius, diameter and distance of a graph C, the effects on the parameter when a vertex or edge is removed or an edge is added, or C is replaced by a spanning tree in which the parameter is as low as possible. We find the maximum possible change in the parameter due to such modifications. In addition, we consider the cases where the removed vertex or edge is one for which the parameter is minimised after deletion. We also investigate graphs which are critical with respect to the radius or diameter, in any of the following senses: the parameter increases when any edge is deleted, decreases when any edge is added, increases when any vertex is removed, or decreases when any vertex is removed
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