共查询到20条相似文献,搜索用时 750 毫秒
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Nicolas Lichiardopol 《Discrete Mathematics》2012,312(12-13):1927-1930
The Bermond–Thomassen conjecture states for , any digraph of minimum out-degree at least contains at least vertex-disjoint directed cycles. In a recent paper, Bessy, Sereni and the author proved that a regular tournament of degree contains at least vertex-disjoint directed cycles, which shows that the above conjecture is true for regular tournaments. In this paper, we improve this result by proving that such a tournament contains at least vertex-disjoint directed cycles. 相似文献
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Aysel Erey 《Discrete Mathematics》2018,341(5):1419-1431
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Let P be a set of n points in . The 2-center problem for P is to find two congruent balls of minimum radius whose union covers P. We present a randomized algorithm for computing a 2-center of P that runs in expected time; here , is the radius of the 2-center balls of P, and is the radius of the smallest enclosing ball of P. The algorithm is near quadratic as long as is not too close to , which is equivalent to the condition that the centers of the two covering balls be not too close to each other. This improves an earlier slightly super-cubic algorithm of Agarwal, Efrat, and Sharir (2000) [2] (at the cost of making the algorithm performance depend on the center separation of the covering balls). 相似文献
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Guoyou Qian 《Comptes Rendus Mathematique》2017,355(11):1127-1132
Let be a strictly increasing sequence of positive integers ( if ). In 1978, Borwein showed that for any positive integer n, we have , with equality occurring if and only if for . Let be an integer. In this paper, we investigate the sum and show that for any positive integer n, where is a constant depending on r and n. Further, for any integer , we also give a characterization of the sequence such that the equality holds. 相似文献
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Let q be a positive integer. Recently, Niu and Liu proved that, if , then the product is not a powerful number. In this note, we prove (1) that, for any odd prime power ? and , the product is not a powerful number, and (2) that, for any positive odd integer ?, there exists an integer such that, for any positive integer , the product is not a powerful number. 相似文献
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The Turán number is the maximum number of edges in any -vertex graph that does not contain a subgraph isomorphic to . A wheel is a graph on vertices obtained from a by adding one vertex and making adjacent to all vertices of the . We obtain two exact values for small wheels: Given that is already known, this paper completes the spectrum for all wheels up to 7 vertices. In addition, we present the construction which gives us the lower bound in general case. 相似文献
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A well-known special case of a conjecture attributed to Ryser (actually appeared in the thesis of Henderson (1971)) states that -partite intersecting hypergraphs have transversals of at most vertices. An equivalent form of the conjecture in terms of coloring of complete graphs is formulated in Gyárfás (1977): if the edges of a complete graph are colored with colors then the vertex set of can be covered by at most sets, each forming a connected graph in some color. It turned out that the analogue of the conjecture for hypergraphs can be answered: it was proved in Király (2013) that in every -coloring of the edges of the -uniform complete hypergraph (), the vertex set of can be covered by at most sets, each forming a connected hypergraph in some color.Here we investigate the analogue problem for complete -uniform -partite hypergraphs. An edge coloring of a hypergraph is called spanning if every vertex is incident to edges of every color used in the coloring. We propose the following analogue of Ryser’s conjecture.In every spanning
-coloring of the edges of a complete
-uniform
-partite hypergraph, the vertex set can be covered by at most
sets, each forming a connected hypergraph in some color.We show that the conjecture (if true) is best possible. Our main result is that the conjecture is true for . We also prove a slightly weaker result for , namely that sets, each forming a connected hypergraph in some color, are enough to cover the vertex set.To build a bridge between complete -uniform and complete -uniform -partite hypergraphs, we introduce a new notion. A hypergraph is complete -uniform -partite if it has all -sets that intersect each partite class in at most vertices (where ).Extending our results achieved for , we prove that for any , in every spanning -coloring of the edges of a complete -uniform -partite hypergraph, the vertex set can be covered by at most sets, each forming a connected hypergraph in some color. 相似文献
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An -edge coloring of a graph is a mapping , where is the color assigned to edge . An exact -edge coloring is an -edge coloring such that there exists an with for all . Let be an edge coloring of . We say is rainbow if no two edges in are assigned the same color by . The anti-Ramsey number, , is the smallest integer such that for any exact -edge coloring of there exists a subgraph isomorphic to that is rainbow. In this paper we confirm a conjecture of Fujita, Kaneko, Schiermeyer, and Suzuki that states , where is a matching of size . 相似文献
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Arindam Khan Sudebkumar P. Pal Mridul Aanjaneya Arijit Bishnu Subhas C. Nandy 《Discrete Applied Mathematics》2013,161(10-11):1496-1505
In this paper we study the diffuse reflection diameter and diffuse reflection radius problems for convex-quadrilateralizable polygons. In the usual model of diffuse reflection, a light ray incident at a point on the reflecting surface is reflected from that point in all possible inward directions. A ray reflected from a polygonal edge may graze that reflecting edge but an incident ray cannot graze the reflecting edge. The diffuse reflection diameter of a simple polygon is the minimum number of diffuse reflections that may be needed in the worst case to illuminate any target point from any point light source inside . We show that the diameter is upper bounded by in our usual model of diffuse reflection for convex-quadrilateralizable polygons. To the best of our knowledge, this is the first upper bound on diffuse reflection diameter within a fraction of for such a class of polygons. We also show that the diffuse reflection radius of a convex-quadrilateralizable simple polygon with vertices is at most under the usual model of diffuse reflection. In other words, there exists a point inside such a polygon from which usual diffuse reflections are always sufficient to illuminate the entire polygon. In order to establish these bounds for the usual model, we first show that the diameter and radius are and respectively, for the same class of polygons for a relaxed model of diffuse reflections; in the relaxed model an incident ray is permitted to graze a reflecting edge before turning and reflecting off the same edge at any interior point on that edge. We also show that the worst-case diameter and radius lower bounds of and respectively, are sometimes attained in the usual model, as well as in the relaxed model of diffuse reflection. 相似文献
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Liuquan Wang 《Discrete Mathematics》2018,341(12):3370-3384
Let be the number of -colored generalized Frobenius partitions of . We establish some infinite families of congruences for and modulo arbitrary powers of 3, which refine the results of Kolitsch. For example, for and , we prove that We give two different proofs to the congruences satisfied by . One of the proofs uses a relation between and due to Kolitsch, for which we provide a new proof in this paper. 相似文献