任意长圈覆盖指定的独立的点

An invariant σ2（G） of a graph is defined as follows： σ2（G） ：= min{d（u） ＋ d（v）｜u, v ∈V（G）,uv ∈ E（G）,u ≠ v} is the minimum degree sum of nonadjacent vertices （when G is a complete graph, we define σ2（G） = ∞）. Let k, s be integers with k ≥ 2 and s ≥ 4, G be a graph of order n sufficiently large compared with s and k. We show that if σ2（G） ≥ n ＋ k- 1, then for any set of k independent vertices v1,..., vk, G has k vertex-disjoint cycles C1,..., Ck such that ｜Ci｜ ≤ s and vi ∈ V（Ci） for all 1 ≤ i ≤ k.
The condition of degree sum σs（G） ≥ n ＋ k - 1 is sharp.

Any Long Cycles Covering Specified Independent Vertices

An invariant $\sigma_2(G)$ of a graph is defined as follows: $\sigma_2(G):=\min\{ d(u)+d(v)|u,v\in V(G),uv \not\in E(G), u\neq v\}$ is the minimum degree sum of nonadjacent vertices~(when $G$ is a complete graph, we define $\sigma_2(G)=\infty$). Let $k$, $s$ be integers with $k\geq 2$ and $s\geq 4$, $G$ be a graph of order $n $ sufficiently large compared with $s$ and $k$. We show that if $\sigma_2(G)\geq n+k-1$, then for any set of $k$ independent vertices $v_1,\ldots,v_k$, $G$ has $k$ vertex-disjoint cycles $C_1,\ldots, C_k$ such that $ |C_i|\leq s$ and $v_i\in V(C_i)$ for all $1\leq i \leq k$. The condition of degree sum $\sigma_2(G)\geq n+k-1$ is sharp.