A result on fractional (a,b,k)-critical covered graphs

TL;DR

This paper establishes a neighborhood condition that guarantees a graph is fractional (a,b,k)-critical covered, extending the understanding of fractional factors in graphs and demonstrating the condition's sharpness.

Contribution

It introduces a new neighborhood condition for fractional (a,b,k)-critical covered graphs and proves its optimality.

Findings

Provides a sufficient neighborhood condition for fractional (a,b,k)-critical coverage.

Shows the condition is sharp, meaning it cannot be improved.

Extends the theory of fractional graph factors with new critical coverage results.

Abstract

For a graph $G$, the set of vertices in $G$ is denoted by $V(G)$, and the set of edges in $G$ is denoted by $E(G)$. A fractional $[a,b]$-factor of a graph $G$ is a function $h$ from $E(G)$ to $[0,1]$ satisfying $a\leq d_G^{h}(v)\leq b$ for every vertex $v$ of $G$, where $d_G^{h}(v)=\sum\limits_{e\in E(v)}{h(e)}$ and $E(v)=\{e=uv:u\in V(G)\}$. A graph $G$ is called fractional $[a,b]$-covered if $G$ contains a fractional $[a,b]$-factor $h$ with $h(e)=1$ for any edge $e$ of $G$. A graph $G$ is called fractional $(a,b,k)$-critical covered if $G-Q$ is fractional $[a,b]$-covered for any $Q\subseteq V(G)$ with $|Q|=k$. In this article, we demonstrate a neighborhood condition for a graph to be fractional $(a,b,k)$-critical covered. Furthermore, we claim that the result is sharp.