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

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 prove the following two results: (i) If s1,s2≥2 are integers and 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,s2≥2 are integers and G is a triangle-free graph of order at least 3 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 result, we also give some corollaries concerning degree conditions for the existence of k vertex-disjoint cycles.