Global cycle properties of locally isometric graphs

Borchert, Adam; Nicol, Skylar; Oellermann, Ortrud R.

doi:10.1016/j.dam.2016.01.026

Cited by 5 publications

(4 citation statements)

References 17 publications

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“…In the first case we can show as before that v i k +1 ∼ v t−1 and in the second case v i k −1 ∼ v 1 . In either case we obtain, as before, a contradiction to Lemma 3.2 (6)…”

Section: (I) Supposesupporting

confidence: 57%

“…Moreover, there does not appear to be an easy way of recognizing which locally hamiltonian graphs are in fact hamiltonian. The class of 'locally isometric graphs' introduced in [6], is a class of graphs satisfying another such local condition. A subgraph…”

Section: Introductionmentioning

confidence: 99%

“…by the open neighbourhood of every vertex in G is an isometric subgraph of G. It was shown in [6] that the problem of deciding whether a locally isometric graph is hamiltonian is NP-complete for graphs with maximum degree at most 8. Locally connected graphs that are sufficiently 'locally dense' were introduced in [7].…”

Section: Introductionmentioning

confidence: 99%

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Global properties of graphs with local degree conditions

Kubicka,

Kubicki,

Oellermann

2015

Preprint

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Let P be a graph property. A graph G is said to be locally P (closed locally P, respectively) if the subgraph induced by the open neighbourhood (closed neighbourhood, respectively) of every vertex in G has property P. A graph G of order n is said to satisfy Dirac's condition if δ(G) ≥ n/2 and it satisfies Ore's condition if for every pair u, v of non-adjacent vertices in G, deg(u) + deg(v) ≥ n. A graph is locally Dirac (locally Ore, respectively) if the subgraph induced by the open neighbourhood of every vertex satisfies Dirac's condition (Ore's condition, respectively). In this paper we establish global properties for graphs that are locally Dirac and locally Ore. In particular we show that these graphs, of sufficiently large order, are 3-connected. For locally Dirac graphs it is shown that the edge connectivity equals the minimum degree and it is illustrated that this results does not extend to locally Ore graphs. We show that ⌊n/3⌋ − 1 is a sharp upper bound on the diameter of every locally Dirac graph of order n. We show that there exist infinite families of planar closed locally Dirac graphs. In contrast, locally Dirac graphs of sufficiently large order are shown to be non-planar. It is known that every closed locally Ore graph is hamiltonian. We show that locally Dirac graphs have an even richer cycle structure by showing that all locally Dirac graphs with maximum degree 11 are in fact fully cycle extendable. This result supports Ryjáček's well-known conjecture; which states that every connected, locally connected graph is weakly pancyclic.

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“…In the first case we can show as before that v i k +1 ∼ v t−1 and in the second case v i k −1 ∼ v 1 . In either case we obtain, as before, a contradiction to Lemma 3.2 (6)…”

Section: (I) Supposesupporting

confidence: 57%

Section: Introductionmentioning

confidence: 99%

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Global properties of graphs with local degree conditions

Kubicka,

Kubicki,

Oellermann

2015

Preprint

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“…A graph G is locally isometric if N (v) is an isometric subgraph (distance preserving subgraph) of G for all v ∈ V (G). It was shown in [8] that the Hamilton Cycle Problem is NP-complete even for locally isometric graphs with maximum degree 8. Nevertheless, it was shown in [8] that Ryjáček's conjecture holds for all locally isometric graphs with maximum degree 6 and without true twins (i.e., pairs of vertices having the same closed neighbourhood); these graphs are in fact fully cycle extendable.…”

Section: Introductionmentioning

confidence: 99%

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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Some of My Favourite Conjectures: Local Conditions Implying Global Cycle Properties

Oellermann

2018

Graph Theory

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Global cycle properties of locally isometric graphs

Cited by 5 publications

References 17 publications

Global properties of graphs with local degree conditions

Global properties of graphs with local degree conditions

Global cycle properties in graphs with large minimum clustering coefficient

Some of My Favourite Conjectures: Local Conditions Implying Global Cycle Properties

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