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《Discrete Mathematics》2020,343(10):111996
A Gallai coloring of a complete graph is an edge coloring without triangles colored with three different colors. A sequence of positive integers is an -sequence if . An -sequence is a G-sequence if there is a Gallai coloring of with colors such that there are edges of color for all . Gyárfás, Pálvölgyi, Patkós and Wales proved that for any integer there exists an integer such that every -sequence is a G-sequence if and only if . They showed that and .We show that and give almost matching lower and upper bounds for by showing that with suitable constants , for all sufficiently large . 相似文献
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In this paper, we study the long-time behavior of solutions of a reaction–diffusion model in a one-dimensional river network, where the river network has two branches, and the water flow speeds in each branch are the same constant . We show the existence of two critical values and 2 with , and prove that when , the population density in every branch of the river goes to 1 as time goes to infinity; when , then, as time goes to infinity, the population density in every river branch converges to a positive steady state strictly below 1; when , the species will be washed down the stream, and so locally the population density converges to 0. Our result indicates that only if the water-flow speed is suitably small (i.e., ), the species will survive in the long run. 相似文献
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Define a -star to be the complete bipartite graph . In a 2014 article, Hoffman and Roberts prove that a partial -star decomposition of can be embedded in a -star decomposition of where is at most if is odd and if is even. In our work, we offer a straightforward construction for embedding partial -star designs and lower these bounds to and , respectively. 相似文献
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For a positive integer , a graph is -knitted if for each subset of vertices, and every partition of into (disjoint) parts for some , one can find disjoint connected subgraphs such that contains for each . In this article, we show that if the minimum degree of an -vertex graph is at least when , then is -knitted. The minimum degree is sharp. As a corollary, we obtain that -contraction-critical graphs are -connected. 相似文献
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In Korchmáros et al. (2018)one-factorizations of the complete graph are constructed for with any odd prime power such that either or . The arithmetic restriction is due to the fact that the vertices of in the construction are the points of a conic in the finite plane of order . Here we work on the Euclidean plane and describe an analogous construction where the role of is taken by a regular -gon. This allows us to remove the above constraints and construct one-factorizations of for every even . 相似文献
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A decomposition of a multigraph is a partition of its edges into subgraphs . It is called an -factorization if every is -regular and spanning. If is a subgraph of , a decomposition of is said to be enclosed in a decomposition of if, for every , is a subgraph of .Feghali and Johnson gave necessary and sufficient conditions for a given decomposition of to be enclosed in some 2-edge-connected -factorization of for some range of values for the parameters , , , , : , and either , or and and , or and . We generalize their result to every and . We also give some sufficient conditions for enclosing a given decomposition of in some 2-edge-connected -factorization of for every and , where is a constant that depends only on , and . 相似文献
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《Discrete Mathematics》2020,343(12):112127
Let be a positive integer. The Bermond–Thomassen conjecture states that, a digraph of minimum out-degree at least contains vertex-disjoint directed cycles. A digraph is called a local tournament if for every vertex of , both the out-neighbours and the in-neighbours of induce tournaments. Note that tournaments form the subclass of local tournaments. In this paper, we verify that the Bermond–Thomassen conjecture holds for local tournaments. In particular, we prove that every local tournament with contains disjoint cycles , satisfying that either has the length at most 4 or is a shortest cycle of the original digraph of for . 相似文献