Disjoint Cycles with Chords in Graphs

Abstract

Let n(1), n(2),..., n(k) be integers, n = Sigma n(i), n(i) >= 3, and let for each 1 <= i <= 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) >= 2(n - k) - 1 contains k vertex disjoint subgraphs H(1)('), H(2)('),..., 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. (C) 2008 Wiley Periodicals Inc. J Graph Theory 60: 87-98, 2009