Disjoint cycles in graphs with distance degree sum conditions

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 vertex-disjoint cycles and degree sum conditions

This paper considers a degree sum condition sufficient to imply the existence of k vertexdisjoint cycles in a graph G. For an integer t ≥ 1, let σt (G) be the smallest sum of degrees of t independent vertices of G. We prove that if G has order at least 7k+1 and σ4(G) ≥ 8k−3, with k ≥ 2, then G contains k vertex-disjoint cycles. We also show that the degree sum condition on σ4(G) is sharp and co...

Minimum Degree and Disjoint Cycles in Claw-Free Graphs

Minimum degree and disjoint cycles in generalized claw-free graphs

For s ≥ 3 a graph is K1,s-free if it does not contain an induced subgraph isomorphic to K1,s. Cycles in K1,3-free graphs, called clawfree graphs, have beenwell studied. In this paper we extend results on disjoint cycles in claw-free graphs satisfying certain minimum degree conditions to K1,s-free graphs, normally called generalized claw-free graphs. In particular, we prove that if G is K1,s-fre...

Edge-disjoint Hamilton Cycles in Regular Graphs of Large Degree

Theorem 1 implies that if G is a A:-regular graph on n vertices and n ^ 2k, then G contains Wr(n + 3an + 2)] edge-disjoint Hamilton cycles. Thus we are able to increase the bound on the number of edge-disjoint Hamilton cycles by adding a regularity condition. In [5] Nash-Williams conjectured that if G satisfies the conditions of Theorem 2, then G contains [i(n + l)] edge-disjoint Hamilton cycle...