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We say a graph is -colorable with of ’s and of ’s if may be partitioned into independent sets and sets whose induced graphs have maximum degree at most . The maximum average degree, , of a graph is the maximum average degree over all subgraphs of . In this note, for nonnegative integers , we show that if , then is -colorable. 相似文献
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Susan A. van Aardt Christoph Brause Alewyn P. Burger Marietjie Frick Arnfried Kemnitz Ingo Schiermeyer 《Discrete Mathematics》2017,340(11):2673-2677
An edge-coloured graph is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a connected graph denoted by , is the smallest number of colours that are needed in order to make properly connected. Our main result is the following: Let be a connected graph of order and . If , then except when and where and 相似文献
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Let be a finite group, written multiplicatively. The Davenport constant of is the smallest positive integer such that every sequence of with elements has a non-empty subsequence with product . Let be the Dihedral Group of order and be the Dicyclic Group of order . Zhuang and Gao (2005) showed that and Bass (2007) showed that . In this paper, we give explicit characterizations of all sequences of such that and is free of subsequences whose product is 1, where is equal to or for some . 相似文献
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Let e be a positive integer, p be an odd prime, , and be the finite field of q elements. Let . The graph is a bipartite graph with vertex partitions and , and edges defined as follows: a vertex is adjacent to a vertex if and only if and . If and , the graph contains no cycles of length less than eight and is edge-transitive. Motivated by certain questions in extremal graph theory and finite geometry, people search for examples of graphs containing no cycles of length less than eight and not isomorphic to the graph , even without requiring them to be edge-transitive. So far, no such graphs have been found. It was conjectured that if both f and g are monomials, then no such graphs exist. In this paper we prove the conjecture. 相似文献
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《Nonlinear Analysis: Real World Applications》2007,8(4):1062-1078
This paper deals with the existence and nonexistence of nonconstant positive steady-state solutions to a ratio-dependent predator–prey model with diffusion and with the homogeneous Neumann boundary condition. We demonstrate that there exists satisfying for , such that if and , then the diffusion can create nonconstant positive steady-state solutions; whereas the diffusion cannot do provided . 相似文献
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Let be a set of at least two vertices in a graph . A subtree of is a -Steiner tree if . Two -Steiner trees and are edge-disjoint (resp. internally vertex-disjoint) if (resp. and ). Let (resp. ) be the maximum number of edge-disjoint (resp. internally vertex-disjoint) -Steiner trees in , and let (resp. ) be the minimum (resp. ) for ranges over all -subset of . Kriesell conjectured that if for any , then . He proved that the conjecture holds for . In this paper, we give a short proof of Kriesell’s Conjecture for , and also show that (resp. ) if (resp. ) in , where . Moreover, we also study the relation between and , where is the line graph of . 相似文献
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Vahan V. Mkrtchyan Samvel S. Petrosyan Gagik N. Vardanyan 《Discrete Mathematics》2010,310(10-11):1588-1613
For and a cubic graph let denote the maximum number of edges that can be covered by matchings. We show that and . Moreover, it turns out that . 相似文献
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