Two‐factors each component of which contains a specified vertex
Yoshimi Egawa, Hikoe Enomoto, Ralph J. Faudree, Hao Li, Ingo Schiermeyer · Journal of Graph Theory · 2003
Abstract 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 {v1,v2,…,vk} ⊂ V(G), there is a 2‐factor of G with precisely k cycles {C1,C2,…,Ck} such that vi ∈ V(Ci) for (1 ≤ i ≤ k) if $n = 3K,\delta(G)\ge\,{{7k-2}\over {3}}$ or 3k + 1 ≤ n ≤ 4k, $\delta (G)\,\ge\,{{2n+k-3}\over {3}}$ or 4k ≤ n ≤ 6k − 3,δ(G) ≥ 3k − 1 or n ≥ 6k − 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