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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

Wei Gao 1,, , Juan Luis García Guirao 2, , Mahmoud Abdel-Aty 3,4,5, and  Wenfei Xi 6,

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.

Keywords: Graph, fractional factor, fractional critical deleted graph, independent set degree condition.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
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
References:
[1]

J. A. Bondy and U. S. R. Mutry, Graph Theory, Springer, Berlin, 2008. doi: 10.1007/978-1-84628-970-5.  Google Scholar

[2]

W. Gao, Some Results on Fractional Deleted Graphs, Doctoral disdertation of Soochow university, 2012. Google Scholar

[3]

W. Gao and Y. Gao, Toughness condition for a graph to be a fractional (g, f, n)-critical deleted graph, The Scientific World Jo., 2014 (2014), Article ID 369798, 7 pages, http://dx.doi.org/10.1155/2014/369798. Google Scholar

[4]

W. Gao, L. Liang, T. W. Xu and J. X. Zhou, Tight toughness condition for fractional (g, f, n)-critical graphs, J. Korean Math. Soc., 51 (2014), 55-65.  doi: 10.4134/JKMS.2014.51.1.055.  Google Scholar

[5]

W. Gao, L. Liang, T. W. Xu and J. X. Zhou, Degree conditions for fractional (g, f, n', m)-critical deleted graphs and fractional ID-(g, f, m)-deleted graphs, Bull. Malays. Math. Sci. Soc., 39 (2016), 315-330.  doi: 10.1007/s40840-015-0194-1.  Google Scholar

[6]

W. Gao and M. R. Farahani, Degree-based indices computation for special chemical molecular structures using edge dividing method, Appl. Math. Nonl. Sc., 1 (2016), 94-117.   Google Scholar

[7]

W. Gao and W. F. Wang, Degree conditions for fractional (k, m)-deleted graphs, Ars. Combin., 113A (2014), 273-285.   Google Scholar

[8]

W. Gao and W. F. Wang, Toughness and fractional critical deleted graph, Utilitas Math., 98 (2015), 295-310.   Google Scholar

[9]

W. Gao and W. F. Wang, A tight neighborhood union condition on fractional (g, f, n, m)-critical deleted graphs, Colloq. Math., 149 (2017), 291-298.  doi: 10.4064/cm6959-8-2016.  Google Scholar

[10]

W. Gao and W. F. Wang, New isolated toughness condition for fractional (g, f, n)-critical graphs, Colloq. Math., 147 (2017), 55-65.  doi: 10.4064/cm6713-8-2016.  Google Scholar

[11]

W. Gao and C. C. Yan, A note on fractional (k, n', m)-critical deleted graph, Advances in Computational Mathematics and its Applications, 1 (2012), 53-55.   Google Scholar

[12]

S. Z. Zhou, A minimum degree condition of fractional (k, m)-deleted graphs, Comptes Rendus Math., 347 (2009), 1223-1226.  doi: 10.1016/j.crma.2009.09.022.  Google Scholar

[13]

S. Z. Zhou, A neighborhood condition for graphs to be fractional (k, m)- deleted graphs, Glasg. Math. J., 52 (2010), 33-40.  doi: 10.1017/S0017089509990139.  Google Scholar

[14]

S. Z. Zhou, A sufficient condition for a graph to be a fractional (f, n)-critical graph, Glasgow Math. J., 52 (2010), 409-415.  doi: 10.1017/S001708951000011X.  Google Scholar

[15]

S. Z. Zhou and H. Liu, On fractional (k, m)-deleted graphs with constrains conditions, Int. J. Comput. Math. Sci., 5 (2011), 130-132.   Google Scholar

[16]

S. Z. Zhou, A sufficient condition for graphs to be fractional (k, m)-deleted graphs, Appl. Math. Lett., 24 (2011), 1533-1538.  doi: 10.1016/j.aml.2011.03.041.  Google Scholar

[17]

S. Z. Zhou and Q. X. Bian, An existence theorem on fractional deleted graphs, Period. Math. Hung., 71 (2015), 125-133.  doi: 10.1007/s10998-015-0089-9.  Google Scholar

show all references

References:
[1]

J. A. Bondy and U. S. R. Mutry, Graph Theory, Springer, Berlin, 2008. doi: 10.1007/978-1-84628-970-5.  Google Scholar

[2]

W. Gao, Some Results on Fractional Deleted Graphs, Doctoral disdertation of Soochow university, 2012. Google Scholar

[3]

W. Gao and Y. Gao, Toughness condition for a graph to be a fractional (g, f, n)-critical deleted graph, The Scientific World Jo., 2014 (2014), Article ID 369798, 7 pages, http://dx.doi.org/10.1155/2014/369798. Google Scholar

[4]

W. Gao, L. Liang, T. W. Xu and J. X. Zhou, Tight toughness condition for fractional (g, f, n)-critical graphs, J. Korean Math. Soc., 51 (2014), 55-65.  doi: 10.4134/JKMS.2014.51.1.055.  Google Scholar

[5]

W. Gao, L. Liang, T. W. Xu and J. X. Zhou, Degree conditions for fractional (g, f, n', m)-critical deleted graphs and fractional ID-(g, f, m)-deleted graphs, Bull. Malays. Math. Sci. Soc., 39 (2016), 315-330.  doi: 10.1007/s40840-015-0194-1.  Google Scholar

[6]

W. Gao and M. R. Farahani, Degree-based indices computation for special chemical molecular structures using edge dividing method, Appl. Math. Nonl. Sc., 1 (2016), 94-117.   Google Scholar

[7]

W. Gao and W. F. Wang, Degree conditions for fractional (k, m)-deleted graphs, Ars. Combin., 113A (2014), 273-285.   Google Scholar

[8]

W. Gao and W. F. Wang, Toughness and fractional critical deleted graph, Utilitas Math., 98 (2015), 295-310.   Google Scholar

[9]

W. Gao and W. F. Wang, A tight neighborhood union condition on fractional (g, f, n, m)-critical deleted graphs, Colloq. Math., 149 (2017), 291-298.  doi: 10.4064/cm6959-8-2016.  Google Scholar

[10]

W. Gao and W. F. Wang, New isolated toughness condition for fractional (g, f, n)-critical graphs, Colloq. Math., 147 (2017), 55-65.  doi: 10.4064/cm6713-8-2016.  Google Scholar

[11]

W. Gao and C. C. Yan, A note on fractional (k, n', m)-critical deleted graph, Advances in Computational Mathematics and its Applications, 1 (2012), 53-55.   Google Scholar

[12]

S. Z. Zhou, A minimum degree condition of fractional (k, m)-deleted graphs, Comptes Rendus Math., 347 (2009), 1223-1226.  doi: 10.1016/j.crma.2009.09.022.  Google Scholar

[13]

S. Z. Zhou, A neighborhood condition for graphs to be fractional (k, m)- deleted graphs, Glasg. Math. J., 52 (2010), 33-40.  doi: 10.1017/S0017089509990139.  Google Scholar

[14]

S. Z. Zhou, A sufficient condition for a graph to be a fractional (f, n)-critical graph, Glasgow Math. J., 52 (2010), 409-415.  doi: 10.1017/S001708951000011X.  Google Scholar

[15]

S. Z. Zhou and H. Liu, On fractional (k, m)-deleted graphs with constrains conditions, Int. J. Comput. Math. Sci., 5 (2011), 130-132.   Google Scholar

[16]

S. Z. Zhou, A sufficient condition for graphs to be fractional (k, m)-deleted graphs, Appl. Math. Lett., 24 (2011), 1533-1538.  doi: 10.1016/j.aml.2011.03.041.  Google Scholar

[17]

S. Z. Zhou and Q. X. Bian, An existence theorem on fractional deleted graphs, Period. Math. Hung., 71 (2015), 125-133.  doi: 10.1007/s10998-015-0089-9.  Google Scholar

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