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 + 1 with σ2(Hi) ≥ 2si − 1. (ii) If s1 and s2 are integers with s1, s2 ≥ 2 and if G is a non-complete triangle-free graph with σ2(G) ≥ 2(s1 + s2)− 1, then G contains two vertex-disjoint subgraphs H1 and H2 such that each Hi is a graph of order at least 2si with σ2(Hi) ≥ 2si−1. By using this kind of results, we also give some corollaries concerning the degree conditions for the existence of vertex-disjoint cycles.