6,782 research outputs found

    On Edge-Disjoint Pairs Of Matchings

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    For a graph G, consider the pairs of edge-disjoint matchings whose union consists of as many edges as possible. Let H be the largest matching among such pairs. Let M be a maximum matching of G. We show that 5/4 is a tight upper bound for |M|/|H|.Comment: 8 pages, 2 figures, Submitted to Discrete Mathematic

    Disjoint compatibility graph of non-crossing matchings of points in convex position

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    Let X2kX_{2k} be a set of 2k2k labeled points in convex position in the plane. We consider geometric non-intersecting straight-line perfect matchings of X2kX_{2k}. Two such matchings, MM and MM', are disjoint compatible if they do not have common edges, and no edge of MM crosses an edge of MM'. Denote by DCMk\mathrm{DCM}_k the graph whose vertices correspond to such matchings, and two vertices are adjacent if and only if the corresponding matchings are disjoint compatible. We show that for each k9k \geq 9, the connected components of DCMk\mathrm{DCM}_k form exactly three isomorphism classes -- namely, there is a certain number of isomorphic small components, a certain number of isomorphic medium components, and one big component. The number and the structure of small and medium components is determined precisely.Comment: 46 pages, 30 figure

    Pairs of disjoint matchings and related classes of graphs

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    For a finite graph GG, we study the maximum 22-edge colorable subgraph problem and a related ratio μ(G)ν(G)\frac{\mu(G)}{\nu(G)}, where ν(G)\nu(G) is the matching number of GG, and μ(G)\mu(G) is the size of the largest matching in any pair (H,H)(H,H') of disjoint matchings maximizing H+H|H| + |H'| (equivalently, forming a maximum 22-edge colorable subgraph). Previously, it was shown that 45μ(G)ν(G)1\frac{4}{5} \le \frac{\mu(G)}{\nu(G)} \le 1, and the class of graphs achieving 45\frac{4}{5} was completely characterized. We show here that any rational number between 45\frac{4}{5} and 11 can be achieved by a connected graph. Furthermore, we prove that every graph with ratio less than 11 must admit special subgraphs

    On disjoint matchings in cubic graphs

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    For i=2,3i=2,3 and a cubic graph GG let νi(G)\nu_{i}(G) denote the maximum number of edges that can be covered by ii matchings. We show that ν2(G)4/5V(G)\nu_{2}(G)\geq {4/5}| V(G)| and ν3(G)7/6V(G)\nu_{3}(G)\geq {7/6}| V(G)| . Moreover, it turns out that ν2(G)V(G)+2ν3(G)4\nu_{2}(G)\leq \frac{|V(G)|+2\nu_{3}(G)}{4}.Comment: 41 pages, 8 figures, minor chage

    Packing Plane Perfect Matchings into a Point Set

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    Given a set PP of nn points in the plane, where nn is even, we consider the following question: How many plane perfect matchings can be packed into PP? We prove that at least log2n2\lceil\log_2{n}\rceil-2 plane perfect matchings can be packed into any point set PP. For some special configurations of point sets, we give the exact answer. We also consider some extensions of this problem

    Rank Maximal Matchings -- Structure and Algorithms

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    Let G = (A U P, E) be a bipartite graph where A denotes a set of agents, P denotes a set of posts and ranks on the edges denote preferences of the agents over posts. A matching M in G is rank-maximal if it matches the maximum number of applicants to their top-rank post, subject to this, the maximum number of applicants to their second rank post and so on. In this paper, we develop a switching graph characterization of rank-maximal matchings, which is a useful tool that encodes all rank-maximal matchings in an instance. The characterization leads to simple and efficient algorithms for several interesting problems. In particular, we give an efficient algorithm to compute the set of rank-maximal pairs in an instance. We show that the problem of counting the number of rank-maximal matchings is #P-Complete and also give an FPRAS for the problem. Finally, we consider the problem of deciding whether a rank-maximal matching is popular among all the rank-maximal matchings in a given instance, and give an efficient algorithm for the problem
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