Degree conditions for 2-factors

Abstract

For any positive integer k, we investigate degree conditions implying that a graph G of order n contains a 2-factor with exactly k components (vertex disjoint cycles). In particular, we prove that for k ≤ (n/4), Ore's classical condition for a graph to be hamiltonian (k = 1) implies that the graph contains a 2-factor with exactly k components. We also obtain a sufficient degree condition for a graph to have k vertex disjoint cycles, at least s of which are 3-cycles and the remaining are 4-cycles for any s ≤ k.

On the Existence of Disjoint Cycles in a Graph

“…In [1], Brandt et al gave the following sufficient conditions to partition a graph into a specified number of vertex-disjoint cycles:…”

Graph partition into K 3s and K 4s

“…Erdös and Faudree [5] conjectured that if G is a graph of order 4k with δ(G) ≥ 2k, then G contains k disjoint quadrilaterals and it is still open. Brandt et al [2] gave a sufficient condition for a graph to have k disjoint cycles which are either triangles or quadrilaterals. They proved that if G is a graph of order n ≥ 3s + 4(k − s) with σ 2 (G) ≥ n + s, s and k are two positive integers with s ≤ k, then G contains k disjoint cycles C 1 , C 2 , .…”

Vertex-disjoint quadrilaterals containing specified edges in a bipartite graph

“…Other corresponding results can be found in [2,3,[6][7][8]11]. In this paper, we investigate the degree conditions that G contains k vertex-disjoint quadrilaterals containing specified edges.…”