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

For a graph $G$, we denote by $\sigma_{2}(G)$ the minimum degree sum of two non-adjacent vertices if $G$ is non-complete; otherwise, $\sigma_{2}(G) = +\infty$. In this paper, we prove the following two results: (i) If $s_{1}, s_{2} \ge 2$ are integers and $G$ is a non-complete graph with $\sigma_{2}(G) \ge 2(s_{1} + s_{2} + 1) - 1$, then $G$ contains two vertex-disjoint subgraphs $H_{1}$ and $H_{2}$ such that each $H_{i}$ is a graph of order at least $s_{i}+1$ with $\sigma_{2}(H_{i}) \ge 2s_{i} - 1$. (ii) If $s_{1}, s_{2} \ge 2$ are integers and $G$ is a triangle-free graph of order at least $3$ with $\sigma_{2}(G) \ge 2(s_{1} + s_{2}) - 1$, then $G$ contains two vertex-disjoint subgraphs $H_{1}$ and $H_{2}$ such that each $H_{i}$ is a graph of order at least $2s_{i}$ with $\sigma_{2}(H_{i}) \ge 2s_{i} - 1$. By using this result, we also give some corollaries concerning degree conditions for the existence of $k$ vertex-disjoint cycles.