Nearly acyclic digraphs
New complexity results for time-constrained dynamical optimal path problems.
New lower bound for multicolor Ramsey numbers for even cycles.
New sufficient conditions for hamiltonian and pancyclic graphs
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.
New upper bounds on the order of cages.
Nonempty intersection of longest paths in a graph with a small matching number
A maximum matching of a graph is a matching of with the largest number of edges. The matching number of a graph , denoted by , is the number of edges in a maximum matching of . 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...
Nonexistence of graphs with cyclic defect.
Non-surjective linear transformations of tropical matrices preserving the cyclicity index
The cyclicity index of a matrix is the cyclicity index of its critical subgraph, namely, the subgraph of the adjacency graph which consists of all cycles of the maximal average weight. The cyclicity index of a graph is the least common multiple of the cyclicity indices of all its maximal strongly connected subgraphs, and the cyclicity index of a strongly connected graph is the least common divisor of the lengths of its (directed) cycles. In this paper we obtain the characterization of linear, possibly...
Note on a Lovász's result
In this paper, we give a generalization of a result of Lovasz from [2].
Note on Gy. Elekes's conjectures concerning unavoidable patterns in proper colorings.
Note on independent sets of a graph
Let the number of -element sets of independent vertices and edges of a graph be denoted by and , respectively. It is shown that the graphs whose every component is a circuit are the only graphs for which the equality is satisfied for all values of .
Note on linear arboricity of graphs with large girth
Note on Petri and Hamiltonian cycles in cubic polyhedral graphs.
Note on Petrie and Hamiltonian cycles in cubic polyhedral graphs
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 -vertex plane cubic graph can be recognized by an -algorithm.
Note on the weight of paths in plane triangulations of minimum degree 4 and 5
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.
Note sur un problème relatif à la marche du cavalier au l'échiquier
Numerical Procedure for Solving a Minimization Eigenvalue Problem.