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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.

Nonempty intersection of longest paths in a graph with a small matching number

Fuyuan Chen (2015)

Czechoslovak Mathematical Journal

A maximum matching of a graph G is a matching of G with the largest number of edges. The matching number of a graph G , denoted by α ' ( G ) , is the number of edges in a maximum matching of G . In 1966, Gallai conjectured that all the longest paths of a connected graph have a common vertex. Although this conjecture has been disproved, finding some nice classes of graphs that support this conjecture is still very meaningful and interesting. In this short note, we prove that Gallai’s conjecture is true for...

Note on a Lovász's result

Dănuţ Marcu (1997)

Mathematica Bohemica

In this paper, we give a generalization of a result of Lovasz from [2].

Note on independent sets of a graph

Jaroslav Ivančo (1994)

Mathematica Bohemica

Let the number of k -element sets of independent vertices and edges of a graph G be denoted by n ( G , k ) and m ( G , k ) , respectively. It is shown that the graphs whose every component is a circuit are the only graphs for which the equality n ( G , k ) = m ( G , k ) is satisfied for all values of k .

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.

Note on the weight of paths in plane triangulations of minimum degree 4 and 5

Tomás Madaras (2000)

Discussiones Mathematicae Graph Theory

The weight of a path in a graph is defined to be the sum of degrees of its vertices in entire graph. It is proved that each plane triangulation of minimum degree 5 contains a path P₅ on 5 vertices of weight at most 29, the bound being precise, and each plane triangulation of minimum degree 4 contains a path P₄ on 4 vertices of weight at most 31.

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