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A note on degree sum conditions for 2-factors with a prescribed number of cycles in bipartite graphs
Let be a balanced bipartite graph of order , and let be the minimum degree sum of two non-adjacent vertices in different partite sets of . In 1963, Moon and Moser proved that if , then is hamiltonian. In this note, we show that if is a positive integer, then the Moon–Moser condition also implies the existence of a 2-factor with exactly cycles for sufficiently large graphs. In order to prove this, we also give a condition for the existence of vertex-disjoint alternating cycles with respect to a chosen perfect matching in . 相似文献
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Let and be two integers with and . For a graph and a vertex of , we use to denote the degree of in . Define to be the minimum value of , where is an independent set of with . This paper proves the following conjecture proposed by Gould et al. (2018). If is a graph of sufficiently large order with , then contains vertex-disjoint cycles. 相似文献
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For bipartite graphs , the bipartite Ramsey number is the least positive integer so that any coloring of the edges of with colors will result in a copy of in the th color for some . In this paper, our main focus will be to bound the following numbers: and for all for and for Furthermore, we will also show that these mentioned bounds are generally better than the bounds obtained by using the best known Zarankiewicz-type result. 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 -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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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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Ryan Alweiss 《Discrete Mathematics》2018,341(4):981-989
The generalized Ramsey number is the smallest positive integer such that any red–blue coloring of the edges of the complete graph either contains a red copy of or a blue copy of . Let denote a cycle of length and denote a wheel with vertices. In 2014, Zhang, Zhang and Chen determined many of the Ramsey numbers of odd cycles versus larger wheels, leaving open the particular case where is even and . They conjectured that for these values of and , . In 2015, Sanhueza-Matamala confirmed this conjecture asymptotically, showing that . In this paper, we prove the conjecture of Zhang, Zhang and Chen for almost all of the remaining cases. In particular, we prove that if , , and . 相似文献
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Jessica De Silva Kristin Heysse Adam Kapilow Anna Schenfisch Michael Young 《Discrete Mathematics》2018,341(2):492-496
For two graphs and , the Turán number is the maximum number of edges in a subgraph of that contains no copy of . Chen, Li, and Tu determined the Turán numbers for all Chen et al. (2009). In this paper we will determine the Turán numbers for all and . 相似文献
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Jeffrey A. Mudrock 《Discrete Mathematics》2018,341(11):3148-3151
DP-coloring (also called correspondence coloring) is a generalization of list coloring recently introduced by Dvo?ák and Postle. Several known bounds for the list chromatic number of a graph , , also hold for the DP-chromatic number of , . On the other hand, there are several properties of the DP-chromatic number that show that it differs with the list chromatic number. In this note we show one such property. It is well known that if and only if . We show that if , and we show that if . 相似文献
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Let be a graph of order . An even squared Hamiltonian cycle (ESHC) of is a Hamiltonian cycle of with chords for all (where for ). When is even, an ESHC contains all bipartite -regular graphs of order . We prove that there is a positive integer such that for every graph of even order , if the minimum degree is , then contains an ESHC. We show that the condition of being even cannot be dropped and the constant cannot be replaced by . Our results can be easily extended to even th powered Hamiltonian cycles for all . 相似文献
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In this paper, we consider combinatorial numbers , mentioned as Catalan triangle numbers where . These numbers unify the entries of the Catalan triangles and for appropriate values of parameters and , i.e., and . In fact, these numbers are suitable rearrangements of the known ballot numbers and some of these numbers are the well-known Catalan numbers that is .We present identities for sums (and alternating sums) of , squares and cubes of and, consequently, for and . In particular, one of these identities solves an open problem posed in Gutiérrez et al. (2008). We also give some identities between and harmonic numbers . Finally, in the last section, new open problems and identities involving are conjectured. 相似文献
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