Factors and Connected Factors in Tough Graphs with High Isolated Toughness

In this paper, we show that every $1$-tough graph with order and isolated toughness at least $r+1$ has a factor whose degrees are $r$, except for at most one vertex with degree $r+1$. Using this result, we conclude that every $3$-tough graph with order and isolated toughness at least $r+1$ has a connected factor whose degrees lie in the set $\{r,r+1\}$, where $r\ge 3$. Also, we show that this factor can be found $m$-tree-connected, when $G$ is a $(2m+\epsilon)$-tough graph with order and isolated toughness at least $r+1$, where $r\ge (2m-1)(2m/\epsilon+1)$ and $\epsilon > 0$. Next, we prove that every $(m+\epsilon)$-tough graph of order at least $2m$ with high enough isolated toughness admits an $m$-tree-connected factor with maximum degree at most $2m+1$. From this result, we derive that every $(2+\epsilon)$-tough graph of order at least three with high enough isolated toughness has a spanning Eulerian subgraph whose degrees lie in the set $\{2,4\}$. In addition, we provide a family of $5/3$-tough graphs with high enough isolated toughness having no connected even factors with bounded maximum degree