1,227 research outputs found

    Matchings, coverings, and Castelnuovo-Mumford regularity

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    We show that the co-chordal cover number of a graph G gives an upper bound for the Castelnuovo-Mumford regularity of the associated edge ideal. Several known combinatorial upper bounds of regularity for edge ideals are then easy consequences of covering results from graph theory, and we derive new upper bounds by looking at additional covering results.Comment: 12 pages; v4 has minor changes for publicatio

    The first order convergence law fails for random perfect graphs

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    We consider first order expressible properties of random perfect graphs. That is, we pick a graph GnG_n uniformly at random from all (labelled) perfect graphs on nn vertices and consider the probability that it satisfies some graph property that can be expressed in the first order language of graphs. We show that there exists such a first order expressible property for which the probability that GnG_n satisfies it does not converge as n→∞n\to\infty.Comment: 11 pages. Minor corrections since last versio

    Combinatorial Problems on HH-graphs

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    Bir\'{o}, Hujter, and Tuza introduced the concept of HH-graphs (1992), intersection graphs of connected subgraphs of a subdivision of a graph HH. They naturally generalize many important classes of graphs, e.g., interval graphs and circular-arc graphs. We continue the study of these graph classes by considering coloring, clique, and isomorphism problems on HH-graphs. We show that for any fixed HH containing a certain 3-node, 6-edge multigraph as a minor that the clique problem is APX-hard on HH-graphs and the isomorphism problem is isomorphism-complete. We also provide positive results on HH-graphs. Namely, when HH is a cactus the clique problem can be solved in polynomial time. Also, when a graph GG has a Helly HH-representation, the clique problem can be solved in polynomial time. Finally, we observe that one can use treewidth techniques to show that both the kk-clique and list kk-coloring problems are FPT on HH-graphs. These FPT results apply more generally to treewidth-bounded graph classes where treewidth is bounded by a function of the clique number
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