关于图中存在不含给定k-因子的[a,b]-因子的度条件

设$1\leq a<b, 0\leq k$是整数. 设$G$是一个含有$k$-因子$Q$且阶为$|G|$的图. 设\delta(G)$表示$G$的最小度, 且$\delta(G)\geq a+k$. 如果$Q$连通, 设$\varepsilon=k$, 否则设$\varepsilon=k+1$.证明:当$b\geq a+\varepsilon-1$时, 如果对$G$的任意两个不相邻的点$x$和$y$都有max$\{d_G(x),d_G(y)\}\geq {\rm max}\{{{a|G|} \over {a+b}},{{(|G|+(a-1)(2a+b+\varepsilon-2))} \over {b+1}}\}+k$, 那么$G$有一个$a, b]$-因子$F$ 使得 $E(F)\cap E(Q)=\emptyset$. 这个度条件是最佳的, 条件$b\geqa+\varepsilon-1$不能去掉. 进一步,得到图存在含给定$k$-因子的$a, b]$-因子的度条件.

DEGREE CONDITIONS FOR GRAPHS TO HAVE [a,b]-FACTORS EXCLUDING A GIVEN k-FACTORS

Let $1\leq a<b, 1\leq k$ be integers. Let $G$ be a graph of order $|G|$ with a $k$-factor $Q$. Suppose that $\delta(G)\geq a+k$ and max$\{d_G(x),d_G(y)\}\geq {\rm max}\{a|G|/(a+b), (|G|+{{(a-1)(2a+b+\varepsilon-2))} \over {b+1}}\}+k$ for each pair of nonadjacent vertices $x$ and $y$ in $G$, where $\varepsilon=k$ if $Q$ is connected, otherwise $\varepsilon=k+1$. Then $G$ has an $a,b]$-factor $F$ such that $E(F)\cap E(Q)=\emptyset$. The lower bound on the degree condition is sharp and the condition $b\geq a+\varepsilon-1$ cannot be deleted. As consequences, the degree conditions are obtained for a graph with a $k$-factor $Q$ to have $a,b]$-factor $F$ such that $E(Q)\subseteq E(F)$.