# The Extension Degree Conditions for Fractional Factor

## TL;DR

This paper extends Gao's graph degree conditions for fractional factors to cases with vertex and edge removals, analyzing the impact of a difference parameter and demonstrating the sharpness of these conditions through counterexamples.

## Contribution

It generalizes Gao's previous results by incorporating vertex and edge removals and the difference between functions, providing new tight degree conditions for fractional factors.

## Key findings

- New degree conditions for fractional factors with vertex/edge removals
- Reformulation of Gao's conditions considering the difference parameter
- Counterexamples demonstrating the sharpness of the conditions

## Abstract

In Gao's previous work, the authors determined several graph degree conditions of a graph which admits fractional factor in particular settings. It was revealed that these degree conditions are tight if $b=f(x)=g(x)=a$ for all vertices $x$ in $G$. In this paper, we continue to discuss these degree conditions for admitting fractional factor in the setting that several vertices and edges are removed and there is a difference $\Delta$ between $g(x)$ and $f(x)$ for every vertex $x$ in $G$. These obtained new degree conditions reformulate Gao's previous conclusions, and show how $\Delta$ acts in the results. Furthermore, counterexamples are structured to reveal the sharpness of degree conditions in the setting $f(x)=g(x)+\Delta$.