1,376 research outputs found

    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 α\alpha of order two such that vv and α(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 (n≥4n\geq4) and the enhanced hypercube Qn,kQ_{n, k} (0≤k≤n−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

    The zero forcing polynomial of a graph

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    Zero forcing is an iterative graph coloring process, where given a set of initially colored vertices, a colored vertex with a single uncolored neighbor causes that neighbor to become colored. A zero forcing set is a set of initially colored vertices which causes the entire graph to eventually become colored. In this paper, we study the counting problem associated with zero forcing. We introduce the zero forcing polynomial of a graph GG of order nn as the polynomial Z(G;x)=∑i=1nz(G;i)xi\mathcal{Z}(G;x)=\sum_{i=1}^n z(G;i) x^i, where z(G;i)z(G;i) is the number of zero forcing sets of GG of size ii. We characterize the extremal coefficients of Z(G;x)\mathcal{Z}(G;x), derive closed form expressions for the zero forcing polynomials of several families of graphs, and explore various structural properties of Z(G;x)\mathcal{Z}(G;x), including multiplicativity, unimodality, and uniqueness.Comment: 23 page

    Average degree conditions forcing a minor

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    Mader first proved that high average degree forces a given graph as a minor. Often motivated by Hadwiger's Conjecture, much research has focused on the average degree required to force a complete graph as a minor. Subsequently, various authors have consider the average degree required to force an arbitrary graph HH as a minor. Here, we strengthen (under certain conditions) a recent result by Reed and Wood, giving better bounds on the average degree required to force an HH-minor when HH is a sparse graph with many high degree vertices. This solves an open problem of Reed and Wood, and also generalises (to within a constant factor) known results when HH is an unbalanced complete bipartite graph

    Fullerenes with the maximum Clar number

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    The Clar number of a fullerene is the maximum number of independent resonant hexagons in the fullerene. It is known that the Clar number of a fullerene with n vertices is bounded above by [n/6]-2. We find that there are no fullerenes whose order n is congruent to 2 modulo 6 attaining this bound. In other words, the Clar number for a fullerene whose order n is congruent to 2 modulo 6 is bounded above by [n/6]-3. Moreover, we show that two experimentally produced fullerenes C80:1 (D5d) and C80:2 (D2) attain this bound. Finally, we present a graph-theoretical characterization for fullerenes, whose order n is congruent to 2 (respectively, 4) modulo 6, achieving the maximum Clar number [n/6]-3 (respectively, [n/6]-2)
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