A neighborhood union condition for fractional $(k,n',m)$-critical deleted graphs

Document Type: Research Paper

Authors

1 Department of Editorial, Yunnan Normal University

2 Department of Applied Mathematics, Iran University of Science and Technology

3 School of Information and Technology, Yunnan Normal University

Abstract

A graph $G$ is called a fractional‎ ‎$(k,n',m)$-critical deleted graph if any $n'$ vertices are removed‎ ‎from $G$ the resulting graph is a fractional $(k,m)$-deleted‎ ‎graph‎. ‎In this paper‎, ‎we prove that for integers $k\ge 2$‎, ‎$n',m\ge0$‎, ‎$n\ge8k+n'+4m-7$‎, ‎and $\delta(G)\ge k+n'+m$‎, ‎if‎ ‎$$|N_{G}(x)\cup N_{G}(y)|\ge\frac{n+n'}{2}$$‎ ‎for each pair of non-adjacent vertices $x$‎, ‎$y$ of $G$‎, ‎then $G$‎ ‎is a fractional $(k,n',m)$-critical deleted graph‎. ‎The bounds for‎ ‎neighborhood union condition‎, ‎the order $n$ and the minimum degree‎ ‎$\delta(G)$ of $G$ are all sharp‎.

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