Vertex-disjoint directed cycles of prescribed length in tournaments with given minimum out-degree and in-degree

Long cycles in graphs with prescribed toughness and minimum degree

A cycle C of a graph G is a D~-cycle if every component of G V(C) has order less than 2. Using the notion of D~-cycles, a number of results are established concerning long cycles in graphs with prescribed toughness and minimum degree. Let G be a t-tough graph on n/> 3 vertices. If 6 > n/(t + 2) + 2 2 for some 2 ~< t + 1, then G contains a Dx-cycle. In particular, if 6>n/(t+ 1)--1, then G is ham...

Diameter Two Graphs of Minimum Order with Given Degree Set

The degree set of a graph is the set of its degrees. Kapoor et al. [Degree sets for graphs, Fund. Math. 95 (1977) 189-194] proved that for every set of positive integers, there exists a graph of diameter at most two and radius one with that degree set. Furthermore, the minimum order of such a graph is determined. A graph is 2-self- centered if its radius and diameter are two. In this paper for ...

Degree sum conditions and vertex-disjoint cycles in a graph

We consider degree sum conditions and the existence of vertex-disjoint cycles in a graph. In this paper, we prove the following: Suppose that G is a graph of order at least 3k + 2 and σ3(G) ≥ 6k − 2, where k ≥ 2. Then G contains k vertex-disjoint cycles. The degree and order conditions are sharp.

On disjoint directed cycles with prescribed minimum lengths

In this paper, we show that the k-Linkage problem is polynomial-time solvable for digraphs with circumference at most 2. We also show that the directed cycles of length at least 3 have the Erdős-Pósa Property : for every n, there exists an integer tn such that for every digraph D, either D contains n disjoint directed cycles of length at least 3, or there is a set T of tn vertices that meets ev...

Minimum Degree and Disjoint Cycles in Claw-Free Graphs