共查询到20条相似文献,搜索用时 31 毫秒
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We discuss dimension theory in the class of all topological groups. For locally compact topological groups there are many classical results in the literature. Dimension theory for non-locally compact topological groups is mysterious. It is for example unknown whether every connected (hence at least 1-dimensional) Polish group contains a homeomorphic copy of . And it is unknown whether there is a homogeneous metrizable compact space the homeomorphism group of which is 2-dimensional. Other classical open problems are the following ones. Let be a topological group with a countable network. Does it follow that ? The same question if is a compact coset space. We also do not know whether the inequality holds for arbitrary topological groups and which are subgroups of -compact topological groups. The aim of this paper is to discuss such and related problems. But we do not attempt to survey the literature. 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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Marcin Pilipczuk Michał Pilipczuk Riste Škrekovski 《Discrete Applied Mathematics》2012,160(16-17):2484-2490
The well-known conjecture of Vizing on the domination number of Cartesian product graphs claims that for any two graphs and , . We disprove its variations on independent domination number and Barcalkin–German number, i.e. Conjectures 9.6 and 9.2 from the recent survey Bre?ar et al. (2012) [4]. We also give some extensions of the double-projection argument of Clark and Suen (2000) [8], showing that their result can be improved in the case of bounded-degree graphs. Similarly, for rainbow domination number we show for every that , which is closely related to Question 9.9 from the same survey. We also prove that the minimum possible counterexample to Vizing’s conjecture cannot have two neighboring vertices of degree two. 相似文献
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Let and be a sufficiently large real number. In this paper, we prove that, for almost all , the Diophantine inequality is solvable in primes . Moreover, we also investigate the problem of six primes and prove that the Diophantine inequality is solvable in primes for sufficiently large real number . 相似文献
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Boštjan Brešar 《Discrete Mathematics》2017,340(10):2398-2401
A long-standing Vizing’s conjecture asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers; one of the most significant results related to the conjecture is the bound of Clark and Suen, , where stands for the domination number, and is the Cartesian product of graphs and . In this note, we improve this bound by employing the 2-packing number of a graph into the formula, asserting that . The resulting bound is better than that of Clark and Suen whenever is a graph with , and in the case has diameter 2 reads as . 相似文献
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A cycle of order is called a -cycle. A non-induced cycle is called a chorded cycle. Let be an integer with . Then a graph of order is chorded pancyclic if contains a chorded -cycle for every integer with . Cream, Gould and Hirohata (Australas. J. Combin. 67 (2017), 463–469) proved that a graph of order satisfying for every pair of nonadjacent vertices , in is chorded pancyclic unless is either or , the Cartesian product of and . They also conjectured that if is Hamiltonian, we can replace the degree sum condition with the weaker density condition
and still guarantee the same conclusion. In this paper, we prove this conjecture by showing that if a graph of order with contains a -cycle, then contains a chorded -cycle, unless and is either or , Then observing that and are exceptions only for , we further relax the density condition for sufficiently large . 相似文献
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We study the Ramsey number for the 3-uniform loose path of length three, , and colors. We show that , for some explicit constant . 相似文献
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Let be a digraph with vertex set and arc set . Let be three positive integers, and let be a pair of nonnegative integer-valued functions defined on with for all . Let be vertex disjoint -subdigraphs of . In this paper, it is proved that every -digraph has a -factorization -orthogonal to every (). 相似文献
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An orthogonally resolvable matching design OMD is a partition of the edges of the complete graph into matchings of size , called blocks, such that the blocks can be resolved in two different ways. Such a design can be represented as a square array whose cells are either empty or contain a matching of size , where every vertex appears exactly once in each row and column. In this paper we show that an OMD exists if and only if except when and or . 相似文献
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The -power graph of a graph is a graph with the same vertex set as , in that two vertices are adjacent if and only if, there is a path between them in of length at most . A -tree-power graph is the -power graph of a tree, a -leaf-power graph is the subgraph of some -tree-power graph induced by the leaves of the tree.We show that (1) every -tree-power graph has NLC-width at most and clique-width at most , (2) every -leaf-power graph has NLC-width at most and clique-width at most , and (3) every -power graph of a graph of tree-width has NLC-width at most , and clique-width at most . 相似文献