Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs

Darko Dimitrov; Hosam Abdo

doi:10.3934/dcdss.2019045

Article Contents

2019, Volume 12, Issue 4&5: 711-721. Doi: 10.3934/dcdss.2019045 open access

This issue Previous Article Libration points in the restricted three-body problem: Euler angles, existence and stability Next Article A real attractor non admitting a connected feasible open set

Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs

Darko Dimitrov^1, and
Hosam Abdo^2,

1.
Faculty of Information Studies, Novo Mesto, Slovenia
2.
Institut für Informatik, Freie Universität Berlin, Takustraße, D-4195 Berlin, Germany

Received: July 2017

Revised: December 2017

Published: March 2019

Addendum: Wei Guo was withdraw from the author list for the article

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Abstract

The problem of data transmission in communication network can betransformed into the problem of fractional factor existing in graph theory. Inrecent years, the data transmission problem in the specificnetwork conditions has received a great deal of attention, and itraises new demands to the corresponding mathematical model. Underthis background, many advanced results are presented on fractionalcritical deleted graphs and fractional ID deleted graphs. In thispaper, we determine that $G$ is a fractional$ (g,f,n',m) $-critical deleted graph if$ δ(G)≥\frac{b^{2}(i-1)}{a}+n'+2m $, $ n>\frac{(a+b)(i(a+b)+2m-2)+bn'}{a} $, and

$|N_{G}(x_{1})\cup N_{G}(x_{2})\cup···\cup N_{G}(x_{i})|≥\frac{b(n+n')}{a+b}$

for any independent subset $ \{x_{1},x_{2},..., x_{i}\} $ of $ V(G) $. Furthermore, the independent set neighborhood union condition for a graph to be fractional ID-$ (g,f,m) $-deleted is raised. Some examples will be manifested to show the sharpness of independent set neighborhood union conditions.

Keywords:

Graph,
fractional factor,
fractional $ (g, f, n', m) $-critical deleted graph,
fractional ID-$ (g, f, m) $-deleted graph,
independent set neighborhood union.

Mathematics Subject Classification: Primary: 05C70; Secondary: 68M10.

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References

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