Disjoint cycles with chords in graphs

Let $n_1,n_2,\ldots,n_k$ be integers, $n=\sum n_i$ , $n_i\ge 3$ , and let for each $1\le i\le k$ , $H_i$ be a cycle or a tree on $n_i$ vertices. We prove that every graph G of order at least n with $\sigma_2(G) \ge 2( n-k) -1$ contains k vertex disjoint subgraphs $H_1',H_2',\ldots,H_k'$ , where $H_i'=H_i$ , if $H_i$ is a tree, and $H_i'$ is a cycle with $n_i-3$ chords incident with a common vertex, if $H_i$ is a cycle. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 87–98, 2009