Two‐factors each component of which contains a specified vertex

It is shown that if G is a graph of order n with minimum degree δ( G ), then for any set of k specified vertices { v 1 , v 2 ,…, v k } ⊂ V ( G ), there is a 2‐factor of G with precisely k cycles { C 1 , C 2 ,…, C k } such that v i ∈ V ( C i ) for (1 ≤ i ≤ k ) if $n = 3K,\delta(G)\ge\,{{7k-2}\over {3}}$ or 3 k + 1 ≤ n ≤ 4 k , $\delta (G)\,\ge\,{{2n+k-3}\over {3}}$ or 4 k ≤ n ≤ 6 k − 3,δ( G ) ≥ 3 k − 1 or n ≥ 6 k − 3, $\delta(G)\ge {{n}\over {2}}$ . Examples are described that indicate this result is sharp. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 188–198, 2003