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Given a graph , let be the set of all cycle lengths contained in and let . Let and let be the greatest common divisor of and all the positive pairwise differences of elements in . We prove that if a Hamiltonian graph of order has at least edges, where is an integer such that , then or is exceptional, by which we mean for some . We also discuss cases where is not exceptional, for example when is prime. Moreover, we show that , which if is bipartite implies that , where is the number of edges in . 相似文献
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For a subgraph of , let be the maximum number of vertices of that are pairwise distance at least three in . In this paper, we prove three theorems. Let be a positive integer, and let be a subgraph of an -connected claw-free graph . We prove that if , then either can be covered by a cycle in , or there exists a cycle in such that . This result generalizes the result of Broersma and Lu that has a cycle covering all the vertices of if . We also prove that if , then either can be covered by a path in , or there exists a path in such that . By using the second result, we prove the third result. For a tree , a vertex of with degree one is called a leaf of . For an integer , a tree which has at most leaves is called a -ended tree. We prove that if , then has a -ended tree covering all the vertices of . This result gives a positive answer to the conjecture proposed by Kano et al. (2012). 相似文献
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Given a graph , the Turán function is the maximum number of edges in a graph on vertices that does not contain as a subgraph. Let be integers and let be a graph consisting of triangles and cycles of odd lengths at least 5 which intersect in exactly one common vertex. Erd?s et al. (1995) determined the Turán function and the corresponding extremal graphs. Recently, Hou et al. (2016) determined and the extremal graphs, where the cycles have the same odd length with . In this paper, we further determine and the extremal graphs, where and . Let be the smallest integer such that, for all graphs on vertices, the edge set can be partitioned into at most parts, of which every part either is a single edge or forms a graph isomorphic to . Pikhurko and Sousa conjectured that for and all sufficiently large . Liu and Sousa (2015) verified the conjecture for . In this paper, we further verify Pikhurko and Sousa’s conjecture for with and . 相似文献
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Two graphs are said to be -cospectral (respectively, -cospectral) if they have the same (respectively, signless) Laplacian spectra, and a graph is said to be (respectively, ) if there does not exist other non-isomorphic graph such that and are -cospectral (respectively, -cospectral). Let be the degree sequence of a graph with vertices. In this paper, we prove that except for two exceptions (respectively, the graphs with ), if is -cospectral (respectively, -cospectral) with a connected graph and , then has the same degree sequence as . A spider graph is a unicyclic graph obtained by attaching some paths to a common vertex of the cycle. As an application of our result, we show that every spider graph and its complement graph are both , which extends the corresponding results of Haemers et al. (2008), Liu et al. (2011), Zhang et al. (2009) and Yu et al. (2014). 相似文献
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Michael Tait 《Discrete Mathematics》2018,341(1):104-108
Let denote that any -coloring of contains a monochromatic . The degree Ramsey number of a graph , denoted by , is . We consider degree Ramsey numbers where is a fixed even cycle. Kinnersley, Milans, and West showed that , and Kang and Perarnau showed that . Our main result is that and . Additionally, we substantially improve the lower bound for for general . 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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Let be a 2-regular graph with vertices and assume that has a strong vertex-magic total labeling. It is shown that the four graphs , , and also have a strong vertex-magic total labeling. These theorems follow from a new use of carefully prescribed Kotzig arrays. To illustrate the power of this technique, we show how just three of these arrays, combined with known labelings for smaller 2-regular graphs, immediately provide strong vertex-magic total labelings for 68 different 2-regular graphs of order 49. 相似文献
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Aysel Erey 《Discrete Mathematics》2018,341(5):1419-1431
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Benjamin Braun Hugo Corrales Scott Corry Luis David García Puente Darren Glass Nathan Kaplan Jeremy L. Martin Gregg Musiker Carlos E. Valencia 《Discrete Mathematics》2018,341(10):2949-2963
Let be a finite, connected graph. An arithmetical structure on is a pair of positive integer vectors such that , where is the adjacency matrix of . We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices ). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients , and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles. 相似文献
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Let be a -connected graph of order . In [1], Bondy (1980) considered a degree sum condition for a graph to have a Hamiltonian cycle, say, to be covered by one cycle. He proved that if , then has a Hamiltonian cycle. On the other hand, concerning a degree sum condition for a graph to be covered by two cycles, Enomoto et al. (1995) [4] proved that if and , then can be covered by two cycles. By these results, we conjecture that if , then can be covered by two cycles. In this paper, we prove the case of this conjecture. In fact, we prove a stronger result; if is 2-connected with , then can be covered by two cycles, or belongs to an exceptional class. 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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Let be a connected graph with vertex set and edge set . For a subset of , the Steiner distance of is the minimum size of a connected subgraph whose vertex set contains . For an integer with , the Steiner-Wiener index is . In this paper, we introduce some transformations for trees that do not increase their Steiner -Wiener index for . Using these transformations, we get a sharp lower bound on Steiner -Wiener index for trees with given diameter, and obtain the corresponding extremal graph as well. 相似文献
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Let be a finite connected graph. In this note, we show that the complexity of can be obtained from the partial derivatives at of a determinant in terms of the Bartholdi zeta function of . Moreover, the second order partial derivatives at of this determinant can all be expressed as the linear combination of the Kirchhoff index, the additive degree-Kirchhoff index, and the multiplicative degree-Kirchhoff index of the graph . 相似文献
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Dong Ye 《Discrete Mathematics》2018,341(5):1195-1198
It was conjectured by Mkrtchyan, Petrosyan and Vardanyan that every graph with has a maximum matching such that any two -unsaturated vertices do not share a neighbor. The results obtained in Mkrtchyan et al. (2010), Petrosyan (2014) and Picouleau (2010) leave the conjecture unknown only for -regular graphs with . All counterexamples for -regular graphs given in Petrosyan (2014) have multiple edges. In this paper, we confirm the conjecture for all -regular simple graphs and also -regular multigraphs with . 相似文献