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 vertex-disjoint 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.