共查询到20条相似文献,搜索用时 31 毫秒
1.
A spanning tree of a properly edge-colored complete graph, , is rainbow provided that each of its edges receives a distinct color. In 1996, Brualdi and Hollingsworth conjectured that if is properly -edge-colored, then the edges of can be partitioned into rainbow spanning trees except when . By means of an explicit, constructive approach, in this paper we construct mutually edge-disjoint rainbow spanning trees for any positive value of . Not only are the rainbow trees produced, but also some structure of each rainbow spanning tree is determined in the process. This improves upon best constructive result to date in the literature which produces exactly three rainbow trees. 相似文献
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Ping Sun 《Discrete Mathematics》2012,312(24):3649-3655
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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
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The Ryser Conjecture which states that there is a transversal of size in a Latin square of odd order is equivalent to finding a rainbow matching of size in a properly edge-colored using colors when is odd. Let be the minimum degree of a graph. Wang proposed a more general question to find a function such that every properly edge-colored graph of order contains a rainbow matching of size , which currently has the best bound of by Lo. Babu, Chandran and Vaidyanathan investigated Wang’s question under a stronger color condition. A strongly edge-colored graph is a properly edge-colored graph in which every monochromatic subgraph is an induced matching. Wang, Yan and Yu proved that every strongly edge-colored graph of order at least has a rainbow matching of size . In this note, we extend this result to graphs of order at least . 相似文献
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In this paper, we consider combinatorial numbers , mentioned as Catalan triangle numbers where . These numbers unify the entries of the Catalan triangles and for appropriate values of parameters and , i.e., and . In fact, these numbers are suitable rearrangements of the known ballot numbers and some of these numbers are the well-known Catalan numbers that is .We present identities for sums (and alternating sums) of , squares and cubes of and, consequently, for and . In particular, one of these identities solves an open problem posed in Gutiérrez et al. (2008). We also give some identities between and harmonic numbers . Finally, in the last section, new open problems and identities involving are conjectured. 相似文献
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An configuration is a set of points and lines such that each point lies on lines while each line contains points. The configuration is geometric, topological, or combinatorial depending on whether lines are considered to be straight lines, pseudolines, or just combinatorial lines. The existence and enumeration of configurations for a given has been subject to active research. A current front of research concerns geometric configurations: it is now known that geometric configurations exist for all , apart from sporadic exceptional cases. In this paper, we settle by computational techniques the first open case of configurations: we obtain all topological configurations among which none are geometrically realizable. 相似文献
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Bart Litjens 《Discrete Mathematics》2018,341(6):1740-1748
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Let be a -connected graph of order . In [1], Bondy (1980) considered a degree sum condition for a graph to have a Hamiltonian cycle, say, to be covered by one cycle. He proved that if , then has a Hamiltonian cycle. On the other hand, concerning a degree sum condition for a graph to be covered by two cycles, Enomoto et al. (1995) [4] proved that if and , then can be covered by two cycles. By these results, we conjecture that if , then can be covered by two cycles. In this paper, we prove the case of this conjecture. In fact, we prove a stronger result; if is 2-connected with , then can be covered by two cycles, or belongs to an exceptional class. 相似文献
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TextFor any given two positive integers and , and any set A of nonnegative integers, let denote the number of solutions of the equation with . In this paper, we determine all pairs of positive integers for which there exists a set such that for all . We also pose several problems for further research.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=EnezEsJl0OY. 相似文献
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Ryan Alweiss 《Discrete Mathematics》2018,341(4):981-989
The generalized Ramsey number is the smallest positive integer such that any red–blue coloring of the edges of the complete graph either contains a red copy of or a blue copy of . Let denote a cycle of length and denote a wheel with vertices. In 2014, Zhang, Zhang and Chen determined many of the Ramsey numbers of odd cycles versus larger wheels, leaving open the particular case where is even and . They conjectured that for these values of and , . In 2015, Sanhueza-Matamala confirmed this conjecture asymptotically, showing that . In this paper, we prove the conjecture of Zhang, Zhang and Chen for almost all of the remaining cases. In particular, we prove that if , , and . 相似文献
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Byungchan Kim 《Discrete Mathematics》2011,311(10-11):835-840
An overpartition of is a non-increasing sequence of positive integers whose sum is in which the first occurrence of a number may be overlined. In this article, we investigate the arithmetic behavior of modulo powers of , where is the number of overpartition -tuples of . Using a combinatorial argument, we determine modulo . Employing the arithmetic of quadratic forms, we deduce that is almost always divisible by . Finally, with the aid of the theory of modular forms, for a fixed positive integer , we show that is divisible by for almost all . 相似文献
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A graph is packable if it is a subgraph of its complement. The following statement was conjectured by Faudree, Rousseau, Schelp and Schuster in 1981: every non-star graph with girth at least is packable.The conjecture was proved by Faudree et al. with the additional condition that has at most edges. In this paper, for each integer , we prove that every non-star graph with girth at least and at most edges is packable, where is for every . This implies that the conjecture is true for sufficiently large planar graphs. 相似文献
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Hossein Hajiabolhassan 《Discrete Mathematics》2011,311(23-24):2663-2668
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《Discrete Mathematics》2007,307(17-18):2209-2216
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Let V be an n-dimensional vector space over the finite field consisting of q elements and let be the Grassmann graph formed by k-dimensional subspaces of V, . Denote by the restriction of to the set of all non-degenerate linear codes. We show that for any two codes the distance in coincides with the distance in only in the case when , i.e. if n is sufficiently large then for some pairs of codes the distances in the graphs and are distinct. We describe one class of such pairs. 相似文献
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Let and denote the maximum degree and the Laplacian spectral radius of a tree , respectively. In this paper we prove that for two trees and on vertices, if and , then , and the bound “” is the best possible. We also prove that for two trees and on vertices with perfect matchings, if and , then . 相似文献
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Ping Sun 《Discrete Mathematics》2018,341(4):1144-1149
This paper considers the enumeration problem of a generalization of standard Young tableau (SYT) of truncated shape. Let be the SYT of shape truncated by whose upper left cell is , where and are partitions of integers. The summation representation of the number of SYT of the truncated shape is derived. Consequently, three closed formulas for SYT of hollow shapes are obtained, including the cases of (i). , (ii). , and (iii). . Finally, an open problem is posed. 相似文献