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A Result on Fractional (a, b, k)-critical Covered Graphs

A Result on Fractional (a, b, k)-critical Covered Graphs A fractional [a, b]-factor of a graph G is a function h from E(G) to [0, 1] satisfying a≤dGh(v)≤b\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a \le d_G^h(v) \le b$$\end{document} for every vertex v of G, where dGh(v)=∑e∈E(v)h(e)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d_G^h(v) = \sum\limits_{e \in E(v)} {h(e)} $$\end{document} and E(v) = {e = uv : u ∈ 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 ⊆ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

A Result on Fractional (a, b, k)-critical Covered Graphs

Acta Mathematicae Applicatae Sinica , Volume 37 (4) – Oct 1, 2021

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Publisher
Springer Journals
Copyright
Copyright © The Editorial Office of AMAS & Springer-Verlag GmbH Germany 2021
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-021-1034-8
Publisher site
See Article on Publisher Site

Abstract

A fractional [a, b]-factor of a graph G is a function h from E(G) to [0, 1] satisfying a≤dGh(v)≤b\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a \le d_G^h(v) \le b$$\end{document} for every vertex v of G, where dGh(v)=∑e∈E(v)h(e)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d_G^h(v) = \sum\limits_{e \in E(v)} {h(e)} $$\end{document} and E(v) = {e = uv : u ∈ 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 ⊆ 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.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Oct 1, 2021

Keywords: graph; neighborhood; fractional [a, b]-factor; fractional [a, b]-covered graph; fractional (a, b, k)-critical covered graph; 05C70; 05C72

References