Simplified existence theorems on all fractional [a,b]-factors

TL;DR

This paper provides simplified theorems for the existence of all fractional [a,b]-factors in large graphs, establishing sharp degree and neighborhood conditions for such factors.

Contribution

It introduces a new characterization for all fractional (g,f)-factors and sharp bounds for the existence of all fractional [a,b]-factors based on degree and neighborhood union conditions.

Findings

Derived a characterization for all fractional (g,f)-factors.

Established sharp degree bounds for all fractional [a,b]-factors.

Proved neighborhood union conditions ensure the existence of all fractional [a,b]-factors.

Abstract

Let $G$ be a graph with order $n$ and let $g, f : V (G)\rightarrow N$ such that $g(v)\leq f(v)$ for all $v\in V(G)$. We say that $G$ has all fractional $(g, f)$-factors if $G$ has a fractional $p$-factor for every $p: V (G)\rightarrow N$ such that $g(v)\leq p(v)\leq f (v)$ for every $v\in V(G)$. Let $a<b$ be two positive integers. %and $G$ \textbf{a graph} of order $n$ sufficiently large %for $a$ and $b$. If $g\equiv a$, $f\equiv b$ and $G$ has all fractional $(g,f)$-factors, then we say that $G$ has all fractional $[a,b]$-factors. Suppose that $n$ is sufficiently large for $a$ and $b$. This paper contains two results on the existence of all $(g,f)$-factors of graphs. First, we derive from Anstee's fractional $(g,f)$-factor theorem a similar characterization for the property of all fractional $(g,f)$-factors. Second, we show that $G$ has all fractional $[a, b]$-factors if the minimum degree is at least $\frac{1}{4a}((a+b-1)^2+4b)$ and every pair of nonadjacent vertices has cardinality of the neighborhood union at least $bn/(a + b)$. These lower bounds are sharp.