### On n-distant Hamiltonian line graphs.

Linda M. Lesniak (1978)

Aequationes mathematicae

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Linda M. Lesniak (1978)

Aequationes mathematicae

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C.ST.J.A. NASH-WILLIAMS (1971)

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Moshe Rosenfeld (1989)

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Branko Grünbaum, J. Malkevitch (1976)

Aequationes mathematicae

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Gary Chartrand, S. F. Kapoor (1974)

Colloquium Mathematicae

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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.

Michal Tkáč, Heinz-Jürgen Voss (2002)

Discussiones Mathematicae Graph Theory

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Magdalena Bojarska (2010)

Discussiones Mathematicae Graph Theory

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We show that every 2-connected (2)-Halin graph is Hamiltonian.

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...

Zdzisław Skupień (1966)

Fundamenta Mathematicae

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Linda M. Lesniak (1977)

Aequationes mathematicae

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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. ...

J.A. Bondy, R. Häggkvist (1981)

Aequationes mathematicae

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