Document Type : Article

**Authors**

^{1}
School of Science, Jiangsu University of Science and Technology Mengxi Road 2, Zhenjiang, Jiangsu 212003, People&#039;s Republic of China

^{2}
School of Science, Jiangsu University of Science and Technology, Mengxi Road 2, Zhenjiang, Jiangsu 212003, P. R. China.

^{3}
Department of Mathematics, Changji University, Changji, Xinjiang 831100, P. R. China.

**Abstract**

A graph G is called a (P≥n, k)-factor critical graph if G − U has a P≥ n -factor for any U ⊆ V(G) with|U|=k. A graphG is called a (P≥n, m)-factor deleted graph if.............

**Keywords**

- Network
- P≥n-factor
- (P≥n, m)-factor
- deleted graph
- (P≥n, k)-factor critical graph
- toughness
- connectivity

**Main Subjects**

References:

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8. Zhou, S., Yang, F., and Sun, Z. "A neighborhood condition for fractional ID-[a; b]-factor-critical graphs", Discussiones Mathematicae Graph Theory, 36(2), pp. 409-418 (2016).

9. Zhou, S., Wu, J., and Zhang, T. "The existence of P3- factor covered graphs", Discussiones Mathematicae Graph Theory, 37(4), pp. 1055-1065 (2017).

10. Zhou, S. "Some results about component factors in graphs", RAIRO-Operations Research, DOI: 10.1051/ro/2017045.

11. Zhou, S., Liu, H., and Zhang, T. "Randomly orthogonal factorizations with constraints in bipartite networks", Chaos, Solitons and Fractals, 112, pp. 31-35 (2018).

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13. Zhou, S. "Remarks on orthogonal factorizations of digraphs", International Journal of Computer Mathematics, 91(10), pp. 2109-2117 (2014).

14. Zhou, S. "A sufficient condition for a graph to be an (a; b; k)-critical graph", International Journal of Computer Mathematics, 87(10), pp. 2202-2211 (2010).

15. Zhou, S. and Sun, Z. "Neighborhood conditions for fractional ID-k-factor-critical graphs", Acta Mathematicae Applicatae Sinica, English Series, 34(3), pp. 636-644 (2018).

16. Kawarabayashi, K., Matsuda, H., Oda, Y., and Ota, K. "Path factors in cubic graphs", Journal of Graph Theory, 39, pp. 188-193 (2002).

17. Razzazi, M. and Sepahvand, A. "Time complexity of two disjoint simple paths", Scientia Iranica, 24(3), pp. 1335-1343 (2017).

18. Johnson, M., Paulusma, D., and Wood, C. "Path factors and parallel knock-out schemes of almost clawfree graphs", Discrete Mathematics, 310, pp. 1413- 1423 (2010).

19. Gao, W., Guirao, J., and Wu, H. "Two tight independent set conditions for fractional (g; f;m)-deleted graphs systems", Qualitative Theory of Dynamical Systems, 17(1), pp. 231-243 (2018).

20. Gao, W. and Wang, W. "Degree sum condition for fractional ID-k-factor-critical graphs", Miskolc Mathematical Notes, 18(2), pp. 751-758 (2017).

21. Gao, W., Liang, L., and Chen, Y. "An isolated toughness condition for graphs to be fractional (k;m)- deleted graphs", Utilitas Mathematica, 105, pp. 303- 316 (2017).

22. Kano, M., Katona, G.Y., and Kiraly, Z. "Packing paths of length at least two", Discrete Mathematics, 283, pp. 129-135 (2004).

23. Kelmans, A. "Packing 3-vertex paths in claw-free graphs and related topics", Discrete Applied Mathematics, 159, pp. 112-127 (2011).

2. Rezaei, S. and Afshin Hemmatyar, A.M. "CMORC: Class-based Multipath On-demand Routing protocol for Cognitive radio networks", Scientia Iranica, 24(6), pp. 3117-3131 (2017).

3. Chvatal, V. "Tough graphs and Hamiltonian circuits", Discrete Mathematics, 5, pp. 215-228 (1973).

4. Bondy, J.A. and Murty, U.S.R., Graph Theory,Springer, Berlin (2008).

5. Wang, H. "Path factors of bipartite graphs", Journal of Graph Theory, 18, pp. 161-167 (1994).

6. Kaneko, A. "A necessary and sufficient condition for the existence of a path factor every component of which is a path of length at least two", Journal of Combinatorial Theory, Series B, 88, pp. 195-218 (2003).

7. Zhou, S. and Zhang, T. "Some existence theorems on all fractional (g; f)-factors with prescribed properties", Acta Mathematicae Applicatae Sinica, English Series, 34(2), pp. 344-350 (2018).

8. Zhou, S., Yang, F., and Sun, Z. "A neighborhood condition for fractional ID-[a; b]-factor-critical graphs", Discussiones Mathematicae Graph Theory, 36(2), pp. 409-418 (2016).

9. Zhou, S., Wu, J., and Zhang, T. "The existence of P3- factor covered graphs", Discussiones Mathematicae Graph Theory, 37(4), pp. 1055-1065 (2017).

10. Zhou, S. "Some results about component factors in graphs", RAIRO-Operations Research, DOI: 10.1051/ro/2017045.

11. Zhou, S., Liu, H., and Zhang, T. "Randomly orthogonal factorizations with constraints in bipartite networks", Chaos, Solitons and Fractals, 112, pp. 31-35 (2018).

12. Zhou, S. and Bian, Q. "Subdigraphs with orthogonal factorizations of digraphs (II)", European Journal of Combinatorics, 36, pp. 198-205 (2014).

13. Zhou, S. "Remarks on orthogonal factorizations of digraphs", International Journal of Computer Mathematics, 91(10), pp. 2109-2117 (2014).

14. Zhou, S. "A sufficient condition for a graph to be an (a; b; k)-critical graph", International Journal of Computer Mathematics, 87(10), pp. 2202-2211 (2010).

15. Zhou, S. and Sun, Z. "Neighborhood conditions for fractional ID-k-factor-critical graphs", Acta Mathematicae Applicatae Sinica, English Series, 34(3), pp. 636-644 (2018).

16. Kawarabayashi, K., Matsuda, H., Oda, Y., and Ota, K. "Path factors in cubic graphs", Journal of Graph Theory, 39, pp. 188-193 (2002).

17. Razzazi, M. and Sepahvand, A. "Time complexity of two disjoint simple paths", Scientia Iranica, 24(3), pp. 1335-1343 (2017).

18. Johnson, M., Paulusma, D., and Wood, C. "Path factors and parallel knock-out schemes of almost clawfree graphs", Discrete Mathematics, 310, pp. 1413- 1423 (2010).

19. Gao, W., Guirao, J., and Wu, H. "Two tight independent set conditions for fractional (g; f;m)-deleted graphs systems", Qualitative Theory of Dynamical Systems, 17(1), pp. 231-243 (2018).

20. Gao, W. and Wang, W. "Degree sum condition for fractional ID-k-factor-critical graphs", Miskolc Mathematical Notes, 18(2), pp. 751-758 (2017).

21. Gao, W., Liang, L., and Chen, Y. "An isolated toughness condition for graphs to be fractional (k;m)- deleted graphs", Utilitas Mathematica, 105, pp. 303- 316 (2017).

22. Kano, M., Katona, G.Y., and Kiraly, Z. "Packing paths of length at least two", Discrete Mathematics, 283, pp. 129-135 (2004).

23. Kelmans, A. "Packing 3-vertex paths in claw-free graphs and related topics", Discrete Applied Mathematics, 159, pp. 112-127 (2011).

Transactions on Computer Science & Engineering and Electrical Engineering (D)

November and December 2019Pages 3510-3514