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《Discrete Mathematics》2022,345(7):112866
Let G be a graph with n vertices. A path decomposition of G is a set of edge-disjoint paths containing all the edges of G. Let denote the minimum number of paths needed in a path decomposition of G. Gallai Conjecture asserts that if G is connected, then . If G is allowed to be disconnected, then the upper bound for was obtained by Donald [7], which was improved to independently by Dean and Kouider [6] and Yan [14]. For graphs consisting of vertex-disjoint triangles, is reached and so this bound is tight. If triangles are forbidden in G, then can be derived from the result of Harding and McGuinness [11], where g denotes the girth of G. In this paper, we also focus on triangle-free graphs and prove that , which improves the above result with . 相似文献
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《Discrete Mathematics》2023,346(1):113151
The Plurality problem - introduced by Aigner - has many variants. In this article we deal with the following version: suppose we are given n balls, each of them colored by one of three colors. A plurality ball is one such that its color class is strictly larger than any other color class. Questioner asks a triplet (or a k-set in general), and Adversary as an answer gives the partition of the triplet (or the k-set) into color classes. Questioner wants to find a plurality ball as soon as possible or show that there is no such ball, while Adversary wants to postpone this.We denote by the largest number of queries needed to ask in the worst case if both play optimally. We provide an almost precise result in the case of even n by proving that for even we have and for odd we haveWe also prove some bounds on the number of queries needed to ask in the case . 相似文献
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Let M be a random rank-r matrix over the binary field , and let be its Hamming weight, that is, the number of nonzero entries of M.We prove that, as with r fixed and tending to a constant, we have that converges in distribution to a standard normal random variable. 相似文献
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In this paper, we consider the following nonlinear elliptic equation involving the fractional Laplacian with critical exponent: where and , is periodic in with . Under some natural conditions on K near a critical point, we prove the existence of multi-bump solutions where the centers of bumps can be placed in some lattices in , including infinite lattices. On the other hand, to obtain positive solution with infinite bumps such that the bumps locate in lattices in , the restriction on is in some sense optimal, since we can show that for , no such solutions exist. 相似文献
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《Discrete Mathematics》2022,345(4):112784
A set S of vertices in a graph G is a dominating set if every vertex not in S is adjacent to a vertex in S. If, in addition, every vertex in S is adjacent to some other vertex in S, then S is a total dominating set. The domination number of G is the minimum cardinality of a dominating set in G, while the total domination number of G is the minimum cardinality of total dominating set in G. A claw-free graph is a graph that does not contain as an induced subgraph. Let G be a connected, claw-free, cubic graph of order n. We show that if we exclude two graphs, then , and this bound is best possible. In order to prove this result, we prove that if we exclude four graphs, then , and this bound is best possible. These bounds improve previously best known results due to Favaron and Henning (2008) [7], Southey and Henning (2010) [19]. 相似文献
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《Discrete Mathematics》2022,345(12):113082
Let G be a graph of order n with an edge-coloring c, and let denote the minimum color-degree of G. A subgraph F of G is called rainbow if all edges of F have pairwise distinct colors. There have been a lot of results on rainbow cycles of edge-colored graphs. In this paper, we show that (i) if , then every vertex of G is contained in a rainbow triangle; (ii) if and , then every vertex of G is contained in a rainbow ; (iii) if G is complete, and , then G contains a rainbow cycle of length at least k, where . 相似文献
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In this paper, we study the existence and concentration behavior of minimizers for , here and where and are constants. By the Gagliardo–Nirenberg inequality, we get the sharp existence of global constraint minimizers of for when , and . For the case , we prove that the global constraint minimizers of behave like for some when c is large, where is, up to translations, the unique positive solution of in and , and . 相似文献
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Compactness of sign-changing solutions to scalar curvature-type equations with bounded negative part
We consider the equation in a closed Riemannian manifold , where , and , . We obtain a sharp compactness result on the sets of sign-changing solutions whose negative part is a priori bounded. We obtain this result under the conditions that and in M, where is the Scalar curvature of the manifold. We show that these conditions are optimal by constructing examples of blowing-up solutions, with arbitrarily large energy, in the case of the round sphere with a constant potential function h. 相似文献