# Factors and connected factors in tough graphs with high isolated toughness

## TL;DR

This paper establishes conditions under which graphs with high isolated toughness contain specific factors and connected factors, extending understanding of graph structure related to toughness and factorization.

## Contribution

It introduces new sufficient conditions for the existence of particular factors and connected factors in graphs with high isolated toughness.

## Key findings

- Graphs satisfying certain isolated toughness conditions have specific factors.
- Conditions for the existence of factors with prescribed degrees are established.
- Connected factors with degrees in a specified set exist under given toughness constraints.

## Abstract

Let $G$ be a graph and let $f$ be a positive integer-valued function on $V(G)$. Assume that for all $S\subseteq V(G)$, $$\sum_{v\in I(G\setminus S)}f(v)(f(v)+1)\le |S|,$$ where $I(G\setminus S)$ denotes the set of isolated vertices of $G\setminus S$. In this paper, we show that if for all $S\subseteq V(G)$, $$\omega(G\setminus S)\le \sum_{v\in S}(f(v)-1)+1,$$ and $\sum_{v\in V(G)}f(v)$ is even, then $G$ has a factor $F$ such that for each vertex $v$, $d_F(v)=f(v)$, where $\omega(G\setminus S)$ denotes the number of components of $G\setminus S$. Moreover, we show that if for all $S\subseteq V(G)$, $$\omega(G\setminus S)\le \frac{1}{4}|S|+1,$$ and $f\ge 2$, then $G$ has a connected factor $H$ such that for each vertex $v$, $d_H(v)\in \{f(v),f(v)+1\}$.