On the existence of vertex-disjoint subgraphs with high degree sum

Vertex-disjoint subgraphs with high degree sums

For a graph G, we denote by σ2(G) the minimum degree sum of two non-adjacent vertices if G is non-complete; otherwise, σ2(G) = +∞. In this paper, we give the following two results; (i) If s1 and s2 are integers with s1, s2 ≥ 2 and if G is a non-complete graph with σ2(G) ≥ 2(s1 + s2 + 1) − 1, then G contains two vertexdisjoint subgraphs H1 and H2 such that each Hi is a graph of order at least si...

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...

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 complete subgraphs of a graph

We conjecture that if G is a graph of order sk, where s ~3 and k 2: 1 are integers, and d(x)+d(y) ~ 2(s-1)k for every pair of non-adjacent vertices x and y of G, then G contains k vertex-disjoint complete subgraphs of order s. This is true when s = 3, [6]. Here we prove this conjecture for k ~ 6.

Disjoint subgraphs of large maximum degree