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Near-homogeneous spherical Latin bitrades

Nicholas J. Cavenagh (2013)

Commentationes Mathematicae Universitatis Carolinae

A planar Eulerian triangulation is a simple plane graph in which each face is a triangle and each vertex has even degree. Such objects are known to be equivalent to spherical Latin bitrades. (A Latin bitrade describes the difference between two Latin squares of the same order.) We give a classification in the near-regular case when each vertex is of degree 4 or 6 (which we call a near-homogeneous spherical Latin bitrade, or NHSLB). The classification demonstrates that any NHSLB is equal to two graphs...

New sufficient conditions for hamiltonian and pancyclic graphs

Ingo Schiermeyer, Mariusz Woźniak (2007)

Discussiones Mathematicae Graph Theory

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.

Note on Petrie and Hamiltonian cycles in cubic polyhedral graphs

Jaroslav Ivančo, Stanislav Jendroľ, Michal Tkáč (1994)

Commentationes Mathematicae Universitatis Carolinae

In this note we show that deciding the existence of a Hamiltonian cycle in a cubic plane graph is equivalent to the problem of the existence of an associated cubic plane multi-3-gonal graph with a Hamiltonian cycle which takes alternately left and right edges at each successive vertex, i.ei̇t is also a Petrie cycle. The Petrie Hamiltonian cycle in an n -vertex plane cubic graph can be recognized by an O ( n ) -algorithm.

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