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1.
Motivated by the relation , holding for the -generalized Catalan numbers of type and , the connection between dominant regions of the -Shi arrangement of type and is investigated. More precisely, it is explicitly shown how copies of the set of dominant regions of the -Shi arrangement of type , biject onto the set of type such regions. This is achieved by exploiting two different viewpoints of the representative alcove of each region: the Shi tableau and the abacus diagram. In the same line of thought, a bijection between copies of the set of -Dyck paths of height
and the set of lattice paths inside an rectangle is provided. 相似文献
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Greg Malen 《Discrete Mathematics》2018,341(9):2567-2574
For any fixed graph , we prove that the topological connectivity of the graph homomorphism complex Hom() is at least , where , for the minimum degree of a vertex in a subgraph . This generalizes a theorem of C?uki? and Kozlov, in which the maximum degree was used in place of , and provides a high-dimensional analogue of the graph theoretic bound for chromatic number, , as . Furthermore, we use this result to examine homological phase transitions in the random polyhedral complexes Hom when for a fixed constant . 相似文献
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Aysel Erey 《Discrete Mathematics》2018,341(5):1419-1431
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A matching in a 3-uniform hypergraph is a set of pairwise disjoint edges. A -matching in a 3-uniform hypergraph is a matching of size . Let be a partition of vertices such that and . Denote by the 3-uniform hypergraph with vertex set consisting of all those edges which contain at least two vertices of . Let be a 3-uniform hypergraph of order such that for any two adjacent vertices . In this paper, we prove contains a -matching if and only if is not a subgraph of . 相似文献
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For a subgraph of , let be the maximum number of vertices of that are pairwise distance at least three in . In this paper, we prove three theorems. Let be a positive integer, and let be a subgraph of an -connected claw-free graph . We prove that if , then either can be covered by a cycle in , or there exists a cycle in such that . This result generalizes the result of Broersma and Lu that has a cycle covering all the vertices of if . We also prove that if , then either can be covered by a path in , or there exists a path in such that . By using the second result, we prove the third result. For a tree , a vertex of with degree one is called a leaf of . For an integer , a tree which has at most leaves is called a -ended tree. We prove that if , then has a -ended tree covering all the vertices of . This result gives a positive answer to the conjecture proposed by Kano et al. (2012). 相似文献
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Francisco Arias Javier de la Cruz Joachim Rosenthal Wolfgang Willems 《Discrete Mathematics》2018,341(10):2729-2734
In this paper we prove that rank metric codes with special properties imply the existence of -analogs of suitable designs. More precisely, we show that the minimum weight vectors of a dually almost MRD code which has no code words of rank weight form a -Steiner system . This is the q-analog of a result in classical coding theory and it may be seen as a first step to prove a q-analog of the famous Assmus–Mattson Theorem. 相似文献
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Let be a prime power and be a positive integer. A subspace partition of , the vector space of dimension over , is a collection of subspaces of such that each nonzero vector of is contained in exactly one subspace in ; the multiset of dimensions of subspaces in is then called a Gaussian partition of . We say that contains a direct sum if there exist subspaces such that . In this paper, we study the problem of classifying the subspace partitions that contain a direct sum. In particular, given integers and with , our main theorem shows that if is a subspace partition of with subspaces of dimension for , then contains a direct sum when has a solution for some integers and belongs to the union of two natural intervals. The lower bound of captures all subspace partitions with dimensions in that are currently known to exist. Moreover, we show the existence of infinite classes of subspace partitions without a direct sum when or when the condition on the existence of a nonnegative integral solution is not satisfied. We further conjecture that this theorem can be extended to any number of distinct dimensions, where the number of subspaces in each dimension has appropriate bounds. These results offer further evidence of the natural combinatorial relationship between Gaussian and integer partitions (when ) as well as subspace and set partitions. 相似文献
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For integers , a -coloring of a graph is a proper coloring with at most colors such that for any vertex with degree , there are at least min different colors present at the neighborhood of . The -hued chromatic number of , , is the least integer such that a -coloring of exists. The list-hued chromatic number of is similarly defined. Thus if , then . We present examples to show that, for any sufficiently large integer , there exist graphs with maximum average degree less than 3 that cannot be -colored. We prove that, for any fraction , there exists an integer such that for each , every graph with maximum average degree is list -colorable. We present examples to show that for some there exist graphs with maximum average degree less than 4 that cannot be -hued colored with less than colors. We prove that, for any sufficiently small real number , there exists an integer such that every graph with maximum average degree satisfies . These results extend former results in Bonamy et al. (2014). 相似文献
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A second order asymptotic expansion in the local limit theorem for a simple branching random walk in
Zhi-Qiang Gao 《Stochastic Processes and their Applications》2018,128(12):4000-4017
Consider a branching random walk, where the underlying branching mechanism is governed by a Galton–Watson process and the migration of particles by a simple random walk in . Denote by the number of particles of generation located at site . We give the second order asymptotic expansion for . The higher order expansion can be derived by using our method here. As a by-product, we give the second order expansion for a simple random walk on , which is used in the proof of the main theorem and is of independent interest. 相似文献
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In this paper, we consider -cycle decomposition of
and directed -cycle decompositions of and , where and denote the wreath product and tensor product of graphs, respectively. Using the results obtained here, we prove that for , the obvious necessary conditions for the existence of a -decomposition of are sufficient whenever where is a prime and . Also, we show that the necessary conditions for the existence of -decompositions of and are sufficient whenever is a prime, where denotes the directed cycle of length . 相似文献
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Tathagata Basak 《Journal of Pure and Applied Algebra》2018,222(10):3036-3042
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Ju Zhou 《Discrete Mathematics》2018,341(4):1021-1031
A graph is induced matching extendable or IM-extendable if every induced matching of is contained in a perfect matching of . In 1998, Yuan proved that a connected IM-extendable graph on vertices has at least edges, and that the only IM-extendable graph with vertices and edges is , where is an arbitrary tree on vertices. In 2005, Zhou and Yuan proved that the only IM-extendable graph with vertices and edges is , where is an arbitrary tree on vertices and is an edge connecting two vertices that lie in different copies of and have distance 3 between them in . In this paper, we introduced the definition of -joint graph and characterized the connected IM-extendable graphs with vertices and edges. 相似文献
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Fares Maalouf 《Journal of Pure and Applied Algebra》2018,222(5):1003-1005
We show that if k is an infinite field, then there exists a subspace of dimension , such that no nonzero member of W has infinitely many zeros. This generalizes a result from a paper by Bergman and Nahlus, and partly answers another question from the same paper. 相似文献
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Let and be two positive integers such that and . A graph is an -parity factor of a graph if is a spanning subgraph of and for all vertices , and . In this paper we prove that every connected graph with vertices has an -parity factor if is even, , and for any two nonadjacent vertices , . This extends an earlier result of Nishimura (1992) and strengthens a result of Cai and Li (1998). 相似文献
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A graph is minimally -tough if the toughness of is and the deletion of any edge from decreases the toughness. Kriesell conjectured that for every minimally -tough graph the minimum degree . We show that in every minimally -tough graph . We also prove that every minimally -tough, claw-free graph is a cycle. On the other hand, we show that for every positive rational number any graph can be embedded as an induced subgraph into a minimally -tough graph. 相似文献