On a Certain Type of Decompositions of Complete Graphs into Factors with Equal Diameters
On a characterization of graphs by average labellings
The additive hereditary property of linear forests is characterized by the existence of average labellings.
On a characterization of -trees
A graph is a -tree if either is the complete graph on vertices, or has a vertex whose neighborhood is a clique of order and the graph obtained by removing from is also a -tree. Clearly, a -tree has at least vertices, and is a 1-tree (usual tree) if and only if it is a -connected graph and has no -minor. In this paper, motivated by some properties of 2-trees, we obtain a characterization of -trees as follows: if is a graph with at least vertices, then is a -tree if...
On a class of polynomially solvable travelling salesman problems
On a class of polynomials associated with the cliques in a graph and its applications.
On a class of polynomials associated with the paths in a graph and its application to minimum nodes disjoint path coverings of graphs.
On a Class of Vertex Folkman Numbers
Let a1 , . . . , ar, be positive integers, i=1 ... r, m = ∑(ai − 1) + 1 and p = max{a1 , . . . , ar }. For a graph G the symbol G → (a1 , . . . , ar ) means that in every r-coloring of the vertices of G there exists a monochromatic ai -clique of color i for some i ∈ {1, . . . , r}. In this paper we consider the vertex Folkman numbers F (a1 , . . . , ar ; m − 1) = min |V (G)| : G → (a1 , . . . , ar ) and Km−1 ⊂ G} We prove that F (a1 , . . . , ar ; m − 1) = m + 6, if p = 3 and m ≧ 6 (Theorem 3)...
On a classification of Hamiltonian tournaments
On a conjecture by Nash-Williams
On a conjecture concerning the Petersen graph.
On a conjecture of Frankl and Füredi.
On a conjecture of M. E. Watkins on graphical regular representations of finite groups
On a conjecture of quintas and arc-traceability in upset tournaments
A digraph D = (V,A) is arc-traceable if for each arc xy in A, xy lies on a directed path containing all the vertices of V, i.e., hamiltonian path. We prove a conjecture of Quintas [7]: if D is arc-traceable, then the condensation of D is a directed path. We show that the converse of this conjecture is false by providing an example of an upset tournament which is not arc-traceable. We then give a characterization for upset tournaments in terms of their score sequences, characterize which arcs of...
On a conjecture of the semigroup of fully indecomposable relations
On a conjecture on the diameter of line graphs of graphs of diameter two
On a connection of number theory with graph theory
We assign to each positive integer a digraph whose set of vertices is and for which there is a directed edge from to if . We establish necessary and sufficient conditions for the existence of isolated fixed points. We also examine when the digraph is semiregular. Moreover, we present simple conditions for the number of components and length of cycles. Two new necessary and sufficient conditions for the compositeness of Fermat numbers are also introduced.
On a discrete Dido-type question.
On a facet of the balanced subgraph polytope
On a family of cubic graphs containing the flower snarks
We consider cubic graphs formed with k ≥ 2 disjoint claws (0 ≤ i ≤ k-1) such that for every integer i modulo k the three vertices of degree 1 of are joined to the three vertices of degree 1 of and joined to the three vertices of degree 1 of . Denote by the vertex of degree 3 of and by T the set . In such a way we construct three distinct graphs, namely FS(1,k), FS(2,k) and FS(3,k). The graph FS(j,k) (j ∈ 1,2,3) is the graph where the set of vertices induce j cycles (note that the graphs...
On a four-colour theorem.