Independence number and connectivity for fractional (a,b,k)-critical covered graphs

TL;DR

This paper establishes new conditions involving independence number and connectivity that guarantee a graph is fractional $(a,b,k)$-critical covered, extending previous degree-based criteria.

Contribution

It introduces independence number and connectivity conditions for fractional $(a,b,k)$-critical covered graphs, broadening the understanding beyond degree conditions.

Findings

Derived a connectivity condition involving $ au(G)$ and $ ext{alpha}(G)$

Verified the sufficiency of the condition for fractional $(a,b,k)$-critical coverage

Extended previous degree-based criteria with new graph invariants

Abstract

A graph $G$ is a fractional $(a,b,k)$-critical covered graph if $G-U$ is a fractional $[a,b]$-covered graph for every $U\subseteq V(G)$ with $|U|=k$, which is first defined by Zhou, Xu and Sun (S. Zhou, Y. Xu, Z. Sun, Degree conditions for fractional $(a,b,k)$-critical covered graphs, Information Processing Letters, DOI: 10.1016/j.ipl.2019.105838). Furthermore, they derived a degree condition for a graph to be a fractional $(a,b,k)$-critical covered graph. In this paper, we gain an independence number and connectivity condition for a graph to be a fractional $(a,b,k)$-critical covered graph and verify that $G$ is a fractional $(a,b,k)$-critical covered graph if $$ \kappa(G)\geq\max\Big\{\frac{2b(a+1)(b+1)+4bk+5}{4b},\frac{(a+1)^{2}\alpha(G)+4bk+5}{4b}\Big\}. $$