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August & September  2019, 12(4&5): 877-886. doi: 10.3934/dcdss.2019058

## An independent set degree condition for fractional critical deleted graphs

 1 School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China 2 Departamento de Matemática Aplicaday Estadística, Universidad Politécnica de Cartagena, Hospital de Marina, 30203-Cartagena, Región de Murcia, Spain 3 Center for Photonics and Smart Materials (CPSM), Zewail City of Science and Technology, Egypt 4 Mathematics Department, Faculty of Sciences, Sohag University, Egypt 5 Communication and Networks Engineering, Gulf University, Kingdom of Bahrain 6 College of Tourism and Geographic Sciences, Yunnan Normal University, Kunming 650500, China

* Corresponding author: Wei Gao(gaowei@ynnu.edu.cn)

Received  November 2017 Revised  January 2018 Published  November 2018

Let $i≥2$, $Δ≥0$, $1≤ a≤ b-Δ$, $n>\frac{(a+b)(ib+2m-2)}{a}+n'$ and $δ(G)≥\frac{b^{2}}{a}+n'+2m$, and let $g,f$ be two integer-valued functions defined on $V(G)$ such that $a≤ g(x)≤ f(x)-Δ≤ b-Δ$ for each $x∈ V(G)$. In this article, it is determined that $G$ is a fractional $(g,f,n',m)$-critical deleted graph if $\max\{d_{1},d_{2},···,d_{i}\}≥\frac{b(n+n')}{a+b}$ for any independent subset $\{x_{1},x_{2},..., x_{i}\}\subseteq V(G)$. The result is tight on independent set degree condition.

Citation: Wei Gao, Juan Luis García Guirao, Mahmoud Abdel-Aty, Wenfei Xi. An independent set degree condition for fractional critical deleted graphs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 877-886. doi: 10.3934/dcdss.2019058
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