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, $\sigma_{2}(G) = +\infty$\mathrm{.}Inthispaper,weprovethefollowingtworesults:(i)If$s_{1}, s_{2} \ge 2$areintegersand$G$isanon-completegraphwith$\sigma_{2}(G) \ge 2(s_{1} + s_{2} + 1) - 1$,then$G$containstwovertex-disjointsubgraphs$H_{1}$and$H_{2}$suchthateach$H_{i}$isagraphoforderatleast$s_{i}+1$with$\sigma_{2}(H_{i}) \ge 2s_{i} - 1$\mathrm{.}(ii)If$s_{1}, s_{2} \ge 2$areintegersand$G$isatriangle-freegraphoforderatleast$3$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.