A degree condition for fractional (<em>g, f, n</em>)-critical covered graphs

Xiangyang Lv; Xiangyang Lv

doi:10.3934/math.2020059

AIMS Mathematics

2020, Volume 5, Issue 2: 872-878. doi: 10.3934/math.2020059

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Research article

A degree condition for fractional (g, f, n)-critical covered graphs

Xiangyang Lv ^,

School of Economics and management, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu 212003, China

Received: 26 September 2019 Accepted: 26 December 2019 Published: 03 January 2020
MSC : 05C70, 90B99

A graph $G$ is called a fractional $(g, f)$-covered graph if for any $e\in E(G)$, $G$ admits a fractional $(g, f)$-factor covering $e$. A graph $G$ is called a fractional $(g, f, n)$-critical covered graph if for any $W\subseteq V(G)$ with $|W| = n$, $G-W$ is a fractional $(g, f)$-covered graph. In this paper, we demonstrate that a graph $G$ of order $p$ is a fractional $(g, f, n)$-critical covered graph if $p\geq\frac{(a+b)(a+b+n+1)-(b-m)n+2}{a+m}$, $\delta(G)\geq\frac{(b-m)(b+1)+2}{a+m}+n$ and for every pair of nonadjacent vertices $u$ and $v$ of $G$, $\max\{d_G(u), d_G(v)\}\geq\frac{(b-m)p+(a+m)n+2}{a+b}$, where $g$ and $f$ are integer-valued functions defined on $V(G)$ satisfying $a\leq g(x)\leq f(x)-m\leq b-m$ for every $x\in V(G)$.
- graph,
- degree condition,
- fractional (g, f)-factor,
- fractional (g, f)-covered graph,
- fractional (g, f, n)-critical covered graph
Citation: Xiangyang Lv. A degree condition for fractional (g, f, n)-critical covered graphs[J]. AIMS Mathematics, 2020, 5(2): 872-878. doi: 10.3934/math.2020059

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Abstract

A graph $G$ is called a fractional $(g, f)$-covered graph if for any $e\in E(G)$, $G$ admits a fractional $(g, f)$-factor covering $e$. A graph $G$ is called a fractional $(g, f, n)$-critical covered graph if for any $W\subseteq V(G)$ with $|W| = n$, $G-W$ is a fractional $(g, f)$-covered graph. In this paper, we demonstrate that a graph $G$ of order $p$ is a fractional $(g, f, n)$-critical covered graph if $p\geq\frac{(a+b)(a+b+n+1)-(b-m)n+2}{a+m}$, $\delta(G)\geq\frac{(b-m)(b+1)+2}{a+m}+n$ and for every pair of nonadjacent vertices $u$ and $v$ of $G$, $\max\{d_G(u), d_G(v)\}\geq\frac{(b-m)p+(a+m)n+2}{a+b}$, where $g$ and $f$ are integer-valued functions defined on $V(G)$ satisfying $a\leq g(x)\leq f(x)-m\leq b-m$ for every $x\in V(G)$.

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