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

Abstract

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.

arXiv comment20 pages