Acta Mathematica Sinica, English Series

, Volume 35, Issue 7, pp 1227–1237

# A Toughness Condition for Fractional (k, m)-deleted Graphs Revisited

• Wei Gao
• Juan L. G. Guirao
• Yao Jun Chen
Article

## Abstract

In computer networks, toughness is an important parameter which is used to measure the vulnerability of the network. Zhou et al. obtains a toughness condition for a graph to be fractional (k, m)-deleted and presents an example to show the sharpness of the toughness bound. In this paper, we remark that the previous example does not work and inspired by this fact, we present a new toughness condition for fractional (k, m)-deleted graphs improving the existing one. Finally, we state an open problem.

## Keywords

Graph fractional factor fractional (k, m)-deleted graph toughness

05C70

## References

1. [1]
Ashwin, P., Postlethwaite, C.: On designing heteroclinic networks from graphs. Physica D, 265, 26–39 (2013)
2. [2]
Basavanagoud, B., Desai, V. R., Patil, S.: (β, α)-Connectivity index of graphs. Applied Mathematics and Nonlinear Sciences, 2, 21–30 (2017)
3. [3]
Bondy, J. A., Mutry, U. S. R.: Graph Theory, Springer, Berlin, 2008
4. [4]
Chvátal, V.: Tough graphs and hamiltonian circuits. Discrete Math., 5, 215–228 (1973)
5. [5]
Crouzeilles, R., Lorini, M. L., Grelle, C. E. D.: Applying graph theory to design networks of protected areas: using inter-patch distance for regional conservation planning. Nat. Conservacao, 9, 219–224 (2011)
6. [6]
de Araujo, D. R. B., Martins, J. F., Bastos, C. J. A.: New graph model to design optical networks. IEEE Commun. Lett., 19, 2130–2133 (2015)
7. [7]
Fardad, M., Lin, F., Jovanovic, M. R.: Design of optimal sparse interconnection graphs for synchronization of oscillator networks. IEEE T. Automat. Contr., 59, 2457–2462 (2014)
8. [8]
Gao, W., Guo, Y., Wang, K. Y.: Ontology algorithm using singular value decomposition and applied in multidisciplinary. Cluster Comput., 19, 2201–2210 (2016)
9. [9]
Gao, W., Wang, W. F.: New isolated toughness condition for fractional (g, f, n)-critical graphs. Colloq. Math., 147, 55–66 (2017)
10. [10]
Gao, W., Wang, W. F.: The eccentric connectivity polynomial of two classes of nanotubes. Chaos Soliton & Fractals, 89, 290–294 (2016)
11. [11]
Gao, W., Wang, W. F.: The fifth geometric arithmetic index of bridge graph and carbon nanocones. J. Differ. Equ. Appl., 23(1–2), 100–109 (2017)
12. [12]
Gao, W., Wang, W. F.: Degree conditions for fractional (k, m)-deleted graphs. Ars Combin., 113A, 273–285 (2014)
13. [13]
Gao, W., Wang, W. F.: A neighborhood union condition for fractional (k, m)-deleted graphs. Ars Combin., 113A, 225–233 (2014)
14. [14]
Guirao, J. L. G., Luo, A. C. J.: New trends in nonlinear dynamics and chaoticity. Nonlinear Dynam., 84, 1–2 (2016)
15. [15]
Haenggi, M., Andrews, J., Baccelli, F., et al.: Stochastic geometry and random graphs for the analysis and design of wireless networks. IEEE J. Sel. Area. Comm., 27, 1025–1028 (2009)
16. [16]
Jin, J. H.: Multiple solutions of the Kirchhoff-type problem in R N. Appl. Math. Nonl. Sc., 1, 229–238 (2016)
17. [17]
Lanzeni, S., Messina, E., Archetti, F.: Graph models and mathematical programming in biochemical network analysis and metabolic engineering design. Comput. Math. Appl., 55, 970–983 (2008)
18. [18]
Liu, G. Z., Zhang, L. J.: Toughness and the existence of fractional k-factors of graphs. Discrete Math., 308, 1741–1748 (2008)
19. [19]
Pishvaee, M. S., Rabbani, M.: A graph theoretic-based heuristic algorithm for responsive supply chain network design with direct and indirect shipment. Adv. Eng. Softw., 42, 57–63 (2011)
20. [20]
Possani, V. N., Callegaro, V., Reis, A. I., et al.: Graph-based transistor network generation method for supergate design. IEEE T. VLSI Syst., 24, 692–705 (2016)
21. [21]
Rahimi, M., Haghighi, A.: A graph portioning approach for hydraulic analysis-design of looped pipe networks. Water Resour. Manag., 29, 5339–5352 (2015)
22. [22]
Rizzelli, G., Tornatore, M., Maier, G., et al.: Impairment-aware design of translucent DWDM networks based on the k-path connectivity graph. J. Opt. Commun. Netw., 4, 356–365 (2012)
23. [23]
Zhou, S.: A neighborhood condition for graphs to be fractional (k, m)-deleted graphs. Glasg. Math. J., 52(1), (2010) 33–40.
24. [24]
Zhou, S. Z., Sun, Z. R., Ye, H.: A toughness condition for fractional (k, m)-deleted graphs. Inform. Process. Lett., 113, 255–259 (2013)
25. [25]
Zhou, S. Z., Xu, L., Xu, Y.: A sufficient condition for the existence of a k-factor excluding a given r-factor. Applied Mathematics and Nonlinear Sciences, 2, 13–20 (2017)
26. [26]
Zhou, S., Yang, F., Sun, Z. R.: A neighborhood condition for fractional ID-[a, b]-factor-critical graphs. Discussiones Mathematicae Graph Theory, 36(2), 409–418 (2016)