论文标题
韧性和与树连接的$ \ {f，f+k \} $的存在 - 因素
Toughness and the existence of tree-connected $\{f,f+k\}$-factors
论文作者
论文摘要
让$ g $为图表，让$ f $是$ v（g）$满足$ 200万美元\ le f \ le b $的正整数值的功能，其中$ b $和$ m $是两个正整数，带有$ b \ ge ge 4m^2 $。在本文中，我们表明，如果$ g $是$ b^2 $ -tough和$ | v（g）| \ ge b^2 $，那么它具有$ m $ - $ - 树连接的因子$ h $，以至于每个顶点$ v $，$ v $，$ v $，$ v $ d_h（v）\ in \ in \ {f（v），f（v），f（v），f（v），$ 1 \ $ \ \ $ sefter after todricize a therforize。树连接因子$ h $，因此对于每个顶点$ v $，$ d_h（v）\ in \ {f（v），f（v）+k \} $。作为一个申请，我们证明，每64b（b-a）^2 $ -togh Graph $ g $的订单至少$ b+1 $，带有$ ab | v（g）| $甚至承认，其学位在$ \ {a，b \} $中的连接因子，$ a $ a $ a $ a $ a $ a $ a $ a $和$ b $ a $ a $ a $ a $ a $ a $ a $ a $ a $ 2 \ l a $ a $ a $ a $ a $ a $ 2 \ a $ a <此外，我们证明，只要$ g $具有$ 2 $ g $，至少有五个。该结果证实了由于Chvátal（1973）而引起的长期猜想的较弱版本。
Let $G$ be a graph and let $f$ be a positive integer-valued function on $V(G)$ satisfying $2m\le f\le b$, where $b$ and $m$ are two positive integers with $b\ge 4m^2$. In this paper, we show that if $G$ is $b^2$-tough and $|V(G)|\ge b^2$, then it has an $m$-tree-connected factor $H$ such that for each vertex $v$, $$d_H(v)\in \{f(v), f(v)+1\}.$$ Next, we generalize this result by giving sufficient conditions for a tough graph to have a tree-connected factors $H$ such that for each vertex $v$, $d_H(v)\in \{f(v), f(v)+k\}$. As an application, we prove that every $64b(b-a)^2$-tough graph $G$ of order at least $b+1$ with $ab|V(G)|$ even admits a connected factor whose degrees lie in the set $\{a,b\}$, where $a$ and $b$ are two integers with $2\le a< b < \frac{6}{5}a$. Moreover, we prove that every $16$-tough graph $G$ of order at least three admits a $2$-connected factor whose degrees lie in the set $\{2,3\}$, provided that $G$ has a $2$-factor with girth at least five. This result confirms a weaker version of a long-standing conjecture due to Chvátal (1973).
