有约束条件的图的(g,f)-因子

设G是一个图,用V(G)和E(G)表示它的顶点集和边集,并设g和f是定义在V(G)上的两个整数值函数且g＜f.图G的一个(g,f)-因子是G的一个支撑子图F使对任意的x∈V(G)有g(x)≤dF(x)≤f(x).如果过图G的任意k条边都有一个(g,f)-因子,则称图G是一个(g,f)-k-覆盖图.如果图G的任意k条边不属于它的一个(g,f)-因子,则称图G是一个(g,f)-k-消去图.作者分别给出了一个图是(g,f)-k-覆盖图和(g,f)-k-消去图的充分条件.

(g, f)-Factors of Graphs with Prescribed Properties

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and let $g$ and $f$ be two integer-valued functions defined on $V(G)$ such that $g<f$ for every $x\in V(G)$. A $(g,f)$-factor of $G$ is a spanning subgraph $F$ of $G$ such that $g(x)\leq d_F(x)\leq f(x)$ for every $x\in V(G)$. A graph $G$ is called a $(g,f)$-$k$-covered graph if every $k$ edge belongs to a $(g,f)$-factor. A graph $G$ is called a $(g,f)$-$k$- deleted graph if every $k$ edges does not belong to a $(g,f)$-factor. A sufficient condition for a graph to be $(g,f)$-$k$-covered and $(g,f)$-$k$-deleted is given, respectively.