Displaying similar documents to “Locally Hamiltonian and planar graphs”

New sufficient conditions for hamiltonian and pancyclic graphs

Ingo Schiermeyer, Mariusz Woźniak (2007)

Discussiones Mathematicae Graph Theory

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For a graph G of order n we consider the unique partition of its vertex set V(G) = A ∪ B with A = {v ∈ V(G): d(v) ≥ n/2} and B = {v ∈ V(G):d(v) < n/2}. Imposing conditions on the vertices of the set B we obtain new sufficient conditions for hamiltonian and pancyclic graphs.

On Uniquely Hamiltonian Claw-Free and Triangle-Free Graphs

Ben Seamone (2015)

Discussiones Mathematicae Graph Theory

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A graph is uniquely Hamiltonian if it contains exactly one Hamiltonian cycle. In this note, we prove that claw-free graphs with minimum degree at least 3 are not uniquely Hamiltonian. We also show that this is best possible by exhibiting uniquely Hamiltonian claw-free graphs with minimum degree 2 and arbitrary maximum degree. Finally, we show that a construction due to Entringer and Swart can be modified to construct triangle-free uniquely Hamiltonian graphs with minimum degree 3. ...

A note on the Song-Zhang theorem for Hamiltonian graphs

Kewen Zhao, Ronald J. Gould (2010)

Colloquium Mathematicae

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An independent set S of a graph G is said to be essential if S has a pair of vertices that are distance two apart in G. In 1994, Song and Zhang proved that if for each independent set S of cardinality k+1, one of the following condition holds: (i) there exist u ≠ v ∈ S such that d(u) + d(v) ≥ n or |N(u) ∩ N(v)| ≥ α (G); (ii) for any distinct u and v in S, |N(u) ∪ N(v)| ≥ n - max{d(x): x ∈ S}, then G is Hamiltonian. We prove that if for each...

On hyper-Zagreb index conditions for hamiltonicity of graphs

Yong Lu, Qiannan Zhou (2022)

Czechoslovak Mathematical Journal

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During the last decade, several research groups have published results on sufficient conditions for the hamiltonicity of graphs by using some topological indices. We mainly study hyper-Zagreb index and some hamiltonian properties. We give some sufficient conditions for graphs to be traceable, hamiltonian or Hamilton-connected in terms of their hyper-Zagreb indices. In addition, we also use the hyper-Zagreb index of the complement of a graph to present a sufficient condition for it to...

Improved Sufficient Conditions for Hamiltonian Properties

Jens-P. Bode, Anika Fricke, Arnfried Kemnitz (2015)

Discussiones Mathematicae Graph Theory

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In 1980 Bondy [2] proved that a (k+s)-connected graph of order n ≥ 3 is traceable (s = −1) or Hamiltonian (s = 0) or Hamiltonian-connected (s = 1) if the degree sum of every set of k+1 pairwise nonadjacent vertices is at least ((k+1)(n+s−1)+1)/2. It is shown in [1] that one can allow exceptional (k+ 1)-sets violating this condition and still implying the considered Hamiltonian property. In this note we generalize this result for s = −1 and s = 0 and graphs that fulfill a certain connectivity...

A conjecture on the prevalence of cubic bridge graphs

Jerzy A. Filar, Michael Haythorpe, Giang T. Nguyen (2010)

Discussiones Mathematicae Graph Theory

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Almost all d-regular graphs are Hamiltonian, for d ≥ 3 [8]. In this note we conjecture that in a similar, yet somewhat different, sense almost all cubic non-Hamiltonian graphs are bridge graphs, and present supporting empirical results for this prevalence of the latter among all connected cubic non-Hamiltonian graphs.

Hamiltonicity in multitriangular graphs

Peter J. Owens, Hansjoachim Walther (1995)

Discussiones Mathematicae Graph Theory

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The family of 5-valent polyhedral graphs whose faces are all triangles or 3s-gons, s ≥ 9, is shown to contain non-hamiltonian graphs and to have a shortness exponent smaller than one.