Degree Conditions for the Existence of Vertex-Disjoint Cycles and Paths: A Survey

Chiba, Shuya; Yamashita, Tomoki

doi:10.1007/s00373-017-1873-5

Cited by 25 publications

(9 citation statements)

References 196 publications

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“…Theorem C supports it by including the case c = 0, since ψ(c) = 2 for the case c = 0. For the case c = 1, related results can be also found in [6,Corollary 3.4.7].…”

Section: Discussionmentioning

confidence: 53%

“…(They actually proved a stronger result, see [1] for the detail. See also [6,Theorem 3.4.16].) Theorem F (Babu and Diwan [1]) Let k and c be positive integers, and let G be a graph of order at least k(c + 3).…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…In order to generalize results on Hamilton cycles, degree conditions for partitioning graphs into a prescribed number of cycles with some additional conditions, have been extensively studied. See a survey paper [6].…”

Section: Introductionmentioning

confidence: 99%

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Partitioning a graph into cycles with a specified number of chords

Chiba

Jiang

Jin

2019

Journal of Graph Theory

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For a graph G, let σ 2 (G) be the minimum degree sum of two non-adjacent vertices in G. A chord of a cycle in a graph G is an edge of G joining two non-consecutive vertices of the cycle. In this paper, we prove the following result, which is an extension of a result of Brandt et al. (J. Graph Theory 24 (1997) 165-173) for large graphs: For positive integers k and c, there exists an integer f (k, c) such that, if G is a graph of order n ≥ f (k, c) and σ 2 (G) ≥ n, then G can be partitioned into k vertex-disjoint cycles, each of which has at least c chords.A graph is hamiltonian if it has a Hamilton cycle, i.e., a cycle containing all the vertices of the graph. It is well known that determining whether a given graph is hamiltonian or not, is NP-complete. Therefore, it is natural to study sufficient conditions for hamiltonicity of graphs.

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“…Theorem C supports it by including the case c = 0, since ψ(c) = 2 for the case c = 0. For the case c = 1, related results can be also found in [6,Corollary 3.4.7].…”

Section: Discussionmentioning

confidence: 53%

Section: Proof Of Theoremmentioning

confidence: 99%

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Partitioning a graph into cycles with a specified number of chords

Chiba

Jiang

Jin

2019

Journal of Graph Theory

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“…Chronologically, in [18,19], the authors independently provided minimum degree sufficient conditions for a graph to be spanning k-cyclable. The problem of spanning k-cyclability of general graphs is well studied in the literature, see the survey article [20,21]. Besides, there are some results on spanning cyclability of special graph families.…”

Section: Conjecture 2 ([17]mentioning

confidence: 99%

Embedding Spanning Disjoint Cycles in Hypercube Networks with Prescribed Edges in Each Cycle

Wu,

Sabir

2023

Axioms

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One of the important issues in evaluating an interconnection network is to study the hamiltonian cycle embedding problems. A graph G is spanning k-edge-cyclable if for any k independent edges e1,e2,…,ek of G, there exist k vertex-disjoint cycles C1,C2,…,Ck in G such that V(C1)∪V(C2)∪⋯∪V(Ck)=V(G) and ei∈E(Ci) for all 1≤i≤k. According to the definition, the problem of finding hamiltonian cycle focuses on k=1. The notion of spanning edge-cyclability can be applied to the problem of identifying faulty links and other related issues in interconnection networks. In this paper, we prove that the n-dimensional hypercube Qn is spanning k-edge-cyclable for 1≤k≤n−1 and n≥2. This is the best possible result, in the sense that the n-dimensional hypercube Qn is not spanning n-edge-cyclable.

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“…The existence of cycles with different constraints in graphs and digraphs have been extensively studied, see [18,87] for cycles of prescribed lengths, [18,29] for vertex-disjoint cycles and [47,61] for edge-disjoint cycles. For more recent work, we refer the reader to [8,9,16,58,60,72].…”

Section: Introductionmentioning

confidence: 99%

Properly colored subgraphs in edge-colored graphs

Han

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Degree Conditions for the Existence of Vertex-Disjoint Cycles and Paths: A Survey

Cited by 25 publications

References 196 publications

Partitioning a graph into cycles with a specified number of chords

Partitioning a graph into cycles with a specified number of chords

Embedding Spanning Disjoint Cycles in Hypercube Networks with Prescribed Edges in Each Cycle

Properly colored subgraphs in edge-colored graphs

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