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The -restricted arc connectivity of digraphs is a common generalization of the arc connectivity and the restricted arc connectivity. An arc subset of a strong digraph is a -restricted arc cut if has a strong component with order at least such that contains a connected subdigraph with order at least . The -restricted arc connectivity of a digraph is the minimum cardinality over all -restricted arc cuts of .Let be a strong digraph with order and minimum degree . In this paper, we first show that exists if and, furthermore, if , where is the minimum 3-degree of . Next, we prove that if . Finally, we give examples showing that these results are best possible in some sense. 相似文献
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Tomáš Vetrík 《Discrete Mathematics》2012,312(2):472-475
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《Applied Mathematics Letters》2005,18(11):1286-1292
First a general model for two-step projection methods is introduced and second it has been applied to the approximation solvability of a system of nonlinear variational inequality problems in a Hilbert space setting. Let be a real Hilbert space and be a nonempty closed convex subset of . For arbitrarily chosen initial points , compute sequences and such that where is a nonlinear mapping on is the projection of onto , and . The two-step model is applied to some variational inequality problems. 相似文献
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《Discrete Mathematics》2007,307(7-8):964-970
The Moore bound for a directed graph of maximum out-degree d and diameter k is . It is known that digraphs of order (Moore digraphs) do not exist for and . Similarly, the Moore bound for an undirected graph of maximum degree d and diameter k is . Undirected Moore graphs only exist in a small number of cases. Mixed (or partially directed) Moore graphs generalize both undirected and directed Moore graphs. In this paper, we shall show that all known mixed Moore graphs of diameter are unique and that mixed Moore graphs of diameter do not exist. 相似文献
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Let be an integer. Bermond and Thomassen conjectured that every digraph with minimum out-degree at least contains vertex-disjoint cycles. Recently Bai, Li and Li proved this conjecture for bipartite digraphs. In this paper we prove that every bipartite tournament with minimum out-degree at least , minimum in-degree at least and partite sets of cardinality at least contains vertex-disjoint 4-cycles whenever . Finally, we show that every bipartite tournament with minimum degree at least contains at least vertex-disjoint 4-cycles. 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
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Daniela Amato 《Discrete Mathematics》2018,341(9):2529-2534
The descendant set of a vertex in a directed graph (digraph) is the subdigraph on the set of vertices reachable by a directed path from . We study the structure of descendant sets in an infinite, primitive, highly arc transitive digraph with out-valency , where is a prime and . It was already known that is a tree when
and we show the same holds when . However, for there are examples of infinite, primitive highly arc transitive digraphs of out-valency whose descendant sets are not trees, for some prime . 相似文献
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Jeong-Hyun Kang 《Discrete Mathematics》2018,341(1):96-103
The vertices of Kneser graph are the subsets of of cardinality , two vertices are adjacent if and only if they are disjoint. The square of a graph is defined on the vertex set of with two vertices adjacent if their distance in is at most 2. Z. Füredi, in 2002, proposed the problem of determining the chromatic number of the square of the Kneser graph. The first non-trivial problem arises when . It is believed that where is a constant, and yet the problem remains open. The best known upper bounds are by Kim and Park: for 1 (Kim and Park, 2014) and for (Kim and Park, 2016). In this paper, we develop a new approach to this coloring problem by employing graph homomorphisms, cartesian products of graphs, and linear congruences integrated with combinatorial arguments. These lead to , where is a constant in , depending on . 相似文献
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Very recently, Thomassé et al. (2017) have given an FPT algorithm for Weighted Independent Set in bull-free graphs parameterized by the weight of the solution, running in time . In this article we improve this running time to . As a byproduct, we also improve the previous Turing-kernel for this problem from to . Furthermore, for the subclass of bull-free graphs without holes of length at most for , we speed up the running time to . As grows, this running time is asymptotically tight in terms of , since we prove that for each integer , Weighted Independent Set cannot be solved in time in the class of -free graphs unless the ETH fails. 相似文献
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We consider nonlinear finite-dimensional scalar-input control systems in the vicinity of an equilibrium.When the linearized system is controllable, the nonlinear system is smoothly small-time locally controllable: whatever and , the state can reach a whole neighborhood of the equilibrium at time T with controls arbitrary small in -norm.When the linearized system is not controllable, we prove that: either the state is constrained to live within a smooth strict manifold, up to a cubic residual, or the quadratic order adds a signed drift with respect to it. This drift holds along a Lie bracket of length , is quantified in terms of an -norm of the control, holds for controls small in -norm and these spaces are optimal. Our proof requires only regularity of the vector field.This work underlines the importance of the norm used in the smallness assumption on the control, even in finite dimension. 相似文献
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Bart Litjens 《Discrete Mathematics》2018,341(6):1740-1748
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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