A sufficient local degree condition for Hamiltonicity in locally finite claw-free graphs

Heuer, Karl

doi:10.1016/j.ejc.2016.01.003

Cited by 15 publications

(18 citation statements)

References 18 publications

(32 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…So we shall show that the following two theorems can be extended to locally finite graphs and affirmatively answer questions of Stein [ We also give some corollaries of the two theorems whose infinite but locally finite counterparts are corollaries to the infinite versions of those theorems. Similar questions on Hamilton circles in infinite graphs were investigated by Heuer [17,18].…”

Section: Introductionmentioning

confidence: 75%

Extending Cycles Locally to Hamilton Cycles

Hamann

Lehner

Pott³

2016

Electron. J. Combin.

View full text Add to dashboard Cite

A Hamilton circle in an infinite graph is a homeomorphic copy of the unit circle $S^1$ that contains all vertices and all ends precisely once. We prove that every connected, locally connected, locally finite, claw-free graph has such a Hamilton circle, extending a result of Oberly and Sumner to infinite graphs. Furthermore, we show that such graphs are Hamilton-connected if and only if they are $3$-connected, extending a result of Asratian. Hamilton-connected means that between any two vertices there is a Hamilton arc, a homeomorphic copy of the unit interval $[0,1]$ that contains all vertices and all ends precisely once.

show abstract

Section: Introductionmentioning

confidence: 75%

Extending Cycles Locally to Hamilton Cycles

Hamann

Lehner

Pott³

2016

Electron. J. Combin.

View full text Add to dashboard Cite

show abstract

“…Several theorems on Hamiltonicity of finite graphs have been extended to Hamilton circles in locally finite graph; for instance, Georgakopoulos [22] extended Fleischner's theorem [21] on the Hamiltonicity of the square of a 2-connected graph, and Heuer [28] and Hamann et al [26] showed that locally finite, claw-free, locally connected graphs have Hamilton circles, extending a result by Oberly and Sumner [35]. Diestel [18] conjectured that Theorem 3.9 could be extended to locally finite graphs, and Heuer [29] proved the conjecture under the additional assumption that the graph is claw-free.…”

Section: Chapter 4 Infinite Graphsmentioning

confidence: 88%

Local Conditions for Cycles in Graphs

Granholm

2019

Linköping Studies in Science and Technology. Licentiate Thesis

View full text Add to dashboard Cite

“…The first one, called a Hamilton circle of G, was introduced by Diestel and and Kühn [19], and the other one, called a Hamiltonian curve of G, was introduced by Kündgen, Li and Thomassen [31] (see the definitions of these two concepts in section 2). Some results on the existence of Hamilton circles in infinite locally finite graphs were obtained in [11,22,[25][26][27] The next result on Hamiltonian curves was proved in [31].…”

Section: Figurementioning

confidence: 96%

Some local–global phenomena in locally finite graphs

Asratian

Granholm

Khachatryan³

2021

Discrete Applied Mathematics

View full text Add to dashboard Cite

In this paper we present some results for a connected infinite graph G with finite degrees where the properties of balls of small radii guarantee the existence of some Hamiltonian and connectivity properties of G. (For a vertex w of a graph G the ball of radius r centered at w is the subgraph of G induced by the set Mr(w) of vertices whose distance from w does not exceed r). In particular, we prove that if every ball of radius 2 in G is 2-connected and G satisfies the condition dG(u) + dG(v) ≥ |M2(w)| − 1 for each path uwv in G, where u and v are non-adjacent vertices, then G has a Hamiltonian curve, introduced by Kündgen, Li and Thomassen (2017). Furthermore, we prove that if every ball of radius 1 in G satisfies Ore's condition (1960) then all balls of any radius in G are Hamiltonian. Finally, we show that the k-connectedness of all balls of radius r in a graph G implies the k-connectedness of all balls in G with radius bigger than r. This is a generalization of a result of Chartrand and Pippert (1974).

show abstract

A sufficient local degree condition for Hamiltonicity in locally finite claw-free graphs

Cited by 15 publications

References 18 publications

Extending Cycles Locally to Hamilton Cycles

Extending Cycles Locally to Hamilton Cycles

Local Conditions for Cycles in Graphs

Some local–global phenomena in locally finite graphs

Contact Info

Product

Resources

About