Global cycle properties in locally connected, locally traceable and locally hamiltonian graphs

Aardt, Susan A. van; Frick, Marietjie; Oellermann, Ortrud R.; Wet, Johan P. de

doi:10.1016/j.dam.2015.09.022

Cited by 12 publications

(32 citation statements)

References 12 publications

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“…As mentioned above these graphs are weakly pancyclic, but infinitely many are nonhamiltonian as was shown in [1]. It was also shown in [1] that if 'local connectedness' is replaced by 'local traceability' in graphs with ∆ = 5, then apart from three exceptional cases all these graphs are fully cycle extendable. With a stronger local condition, namely 'local Hamiltonicity', it was shown in [1] that an even richer cycle structure is guaranteed.…”

Section: Conjecture 1 (Ryjáček) Every Locally Connected Graph Is Weamentioning

confidence: 67%

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Global cycle properties of locally isometric graphs

Borchert

Nicol

Oellermann

2016

Discrete Applied Mathematics

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Let P be a graph property. A graph G is said to be locally P if the subgraph induced by the open neighbourhood of every vertex in G has property P. Ryjáček's well-known conjecture that every connected, locally connected graph is weakly pancyclic motivated us to consider the global cycle structure of locally P graphs, where P is the property of having diameter at most k for some fixed k ≥ 1. For k = 2 these graphs are called locally isometric graphs. For ∆ ≤ 5, we obtain a complete structural characterization of locally isometric graphs that are not fully cycle extendable. For ∆ = 6, it is shown that locally isometric graphs that are not fully cycle extendable contain a pair of true twins of degree 6. Infinite classes of locally isometric graphs with ∆ = 6 that are not fully cycle extendable are described and observations are made that suggest that a complete characterization of these graphs is unlikely. It is shown that Ryjáček's conjecture holds for all locally isometric graphs with ∆ ≤ 6. The Hamilton cycle problem for locally isometric graphs with maximum degree at most 8 is shown to be NP-complete.

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Section: Conjecture 1 (Ryjáček) Every Locally Connected Graph Is Weamentioning

confidence: 67%

“…However, even for graphs with ∆ = 6 this problems is largely unsolved. Weaker forms of this conjecture are considered in [1] where it was shown that locally Hamiltonian graphs with ∆ = 6 are fully cycle extendable.…”

Section: Conjecture 1 (Ryjáček) Every Locally Connected Graph Is Weamentioning

confidence: 99%

Global cycle properties of locally isometric graphs

Borchert

Nicol

Oellermann

2016

Discrete Applied Mathematics

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“…In fact, a graph G is locally traceable if and only if every ball of radius 1 in G is Hamiltonian. Some cyclic properties of locally Hamiltonian and locally traceable graphs were found in [6,14,16,37,38] We call a locally finite graph G uniformly Hamiltonian if every ball of finite radius in G is Hamiltonian. This concept was defined for finite graphs in [6], where some classes of uniformly Hamiltonian graphs were found.…”

Section: Locally Finite Graphs With Hamiltonian Ballsmentioning

confidence: 99%

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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“…Ryjáček's conjecture seems to be very difficult to settle, so it is natural to consider weaker conjectures. This conjecture has been studied, for example, for locally traceable and locally hamiltonian graphs with maximum degree at most 5 and 6, respectively, see [1], neither of which need to be hamiltonian, see [2]. One may well ask whether there are local connectedness conditions that guarantee (global) hamiltonicity.…”

Section: Introductionmentioning

confidence: 99%

“…Since ξ(v 0 ) ≥ 1 2 , it follows that |E( N (v 0 ) )| ≥ 5. Since G is locally connected and no off-cycle neighbour of v 0 is adjacent with v 1 or v t−1 , by Lemma 1 (1), v 0 has either one or two off-cycle neighbours.…”

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confidence: 98%

Global cycle properties in graphs with large minimum clustering coefficient

Borchert

Nicol

Oellermann

2016

Quaestiones Mathematicae

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Let P be a graph property. A graph G is said to be locally P (closed locally P) if the subgraph induced by the open neighbourhood (closed neighbourhood, respectively) of every vertex in G has property P. The clustering coefficient of a vertex is the proportion of pairs of its neighbours that are themselves neighbours. The minimum clustering coefficient of G is the smallest clustering coefficient among all vertices of G. Let H be a subgraph of a graph G and let S ⊆ V (H). We say that H is a strongly induced subgraph of G with attachment set S, if H is an induced subgraph of G and the vertices of V (H) − S are not incident with edges that are not in H. A graph G is fully cycle extendable if every vertex of G lies in a triangle and for every nonhamiltonian cycle C of G, there is a cycle of length |V (C)| + 1 that contains the vertices of C. A complete characterization, of those locally connected graphs with minimum clustering coefficient 1/2 and maximum degree at most 6 that are fully cycle extendable, is given in terms of forbidden strongly induced subgraphs (with specified attachment sets). Moreover, it is shown that all locally connected graphs with ∆ ≤ 6 and sufficiently large minimum clustering coefficient are weakly pancylic, thereby proving Ryjáček's conjecture for this class of graphs.

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Global cycle properties in locally connected, locally traceable and locally hamiltonian graphs

Cited by 12 publications

References 12 publications

Global cycle properties of locally isometric graphs

Global cycle properties of locally isometric graphs

Some local–global phenomena in locally finite graphs

Global cycle properties in graphs with large minimum clustering coefficient

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