A sufficient condition for Hamiltonicity in locally finite graphs

Heuer, Karl

doi:10.1016/j.ejc.2014.08.025

Cited by 17 publications

(25 citation statements)

References 19 publications

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“…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: 78%

Extending Cycles Locally to Hamilton Cycles

Hamann

Lehner

Pott³

2016

Electron. J. Combin.

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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.

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Section: Introductionmentioning

confidence: 78%

Extending Cycles Locally to Hamilton Cycles

Hamann

Lehner

Pott³

2016

Electron. J. Combin.

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“…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: 95%

Some local–global phenomena in locally finite graphs

Asratian

Granholm

Khachatryan³

2021

Discrete Applied Mathematics

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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).

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“…In this section we collect some lemmas which we shall need later for the proof of the main result. The proof of each statement of this section can be found in [14,Section 3]. We begin with two basic facts about minimal vertex separators in claw-free graphs.…”

Section: Toolkitmentioning

confidence: 99%

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

Heuer

2016

European Journal of Combinatorics

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Among the well-known sufficient degree conditions for the Hamiltonicity of a finite graph, the condition of Asratian and Khachatrian is the weakest and thus gives the strongest result. Diestel conjectured that it should extend to locally finite infinite graphs G, in that the same condition implies that the Freudenthal compactification of G contains a circle through all its vertices and ends. We prove Diestel's conjecture for claw-free graphs.

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A sufficient condition for Hamiltonicity in locally finite graphs

Cited by 17 publications

References 19 publications

Extending Cycles Locally to Hamilton Cycles

Extending Cycles Locally to Hamilton Cycles

Some local–global phenomena in locally finite graphs

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

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