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: $sigma_2(G):=min{ d(u)+d(v)|u,vin V(G),uv notin E(G), uneq v}$ 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 $kgeq 2$ and $sgeq 4$, $G$ 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 $v_iin V(C_i)$ for all $1leq i leq k$. The condition of degree sum $sigma_2(G)geq n+k-1$ is sharp.