21,812 research outputs found

    Forcing large tight components in 3-graphs

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    Any nn-vertex 33-graph with minimum codegree at least ⌊n/3βŒ‹\lfloor n/3\rfloor must have a spanning tight component, but immediately below this threshold it is possible for no tight component to span more than ⌈2n/3βŒ‰\lceil 2n/3\rceil vertices. Motivated by this observation, we ask which codegree forces a tight component of at least any given size. The corresponding function seems to have infinitely many discontinuities, but we provide upper and lower bounds, which asymptotically converge as the function nears the origin.Comment: 10 pages. Final version accepted by European J. Combi

    Tight upper bound on the maximum anti-forcing numbers of graphs

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    Let GG be a simple graph with a perfect matching. Deng and Zhang showed that the maximum anti-forcing number of GG is no more than the cyclomatic number. In this paper, we get a novel upper bound on the maximum anti-forcing number of GG and investigate the extremal graphs. If GG has a perfect matching MM whose anti-forcing number attains this upper bound, then we say GG is an extremal graph and MM is a nice perfect matching. We obtain an equivalent condition for the nice perfect matchings of GG and establish a one-to-one correspondence between the nice perfect matchings and the edge-involutions of GG, which are the automorphisms Ξ±\alpha of order two such that vv and Ξ±(v)\alpha(v) are adjacent for every vertex vv. We demonstrate that all extremal graphs can be constructed from K2K_2 by implementing two expansion operations, and GG is extremal if and only if one factor in a Cartesian decomposition of GG is extremal. As examples, we have that all perfect matchings of the complete graph K2nK_{2n} and the complete bipartite graph Kn,nK_{n, n} are nice. Also we show that the hypercube QnQ_n, the folded hypercube FQnFQ_n (nβ‰₯4n\geq4) and the enhanced hypercube Qn,kQ_{n, k} (0≀k≀nβˆ’40\leq k\leq n-4) have exactly nn, n+1n+1 and n+1n+1 nice perfect matchings respectively.Comment: 15 pages, 7 figure

    An approximate version of Sidorenko's conjecture

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    A beautiful conjecture of Erd\H{o}s-Simonovits and Sidorenko states that if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. This conjecture also has an equivalent analytic form and has connections to a broad range of topics, such as matrix theory, Markov chains, graph limits, and quasirandomness. Here we prove the conjecture if H has a vertex complete to the other part, and deduce an approximate version of the conjecture for all H. Furthermore, for a large class of bipartite graphs, we prove a stronger stability result which answers a question of Chung, Graham, and Wilson on quasirandomness for these graphs.Comment: 12 page
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