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On the Problem of Determining which (n, k)-Star Graphs are Cayley Graphs

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Abstract

In this paper we work to classify which of the (n, k)-star graphs, denoted by \(S_{n,k}\), are Cayley graphs. Although the complete classification is left open, we derive infinite and non-trivial classes of both Cayley and non-Cayley graphs. We give a complete classification of the case when \(k=2\), showing that \(S_{n,2}\) is Cayley if and only if n is a prime power. We also give a sufficient condition for \(S_{n,3}\) to be Cayley and study other structural properties, such as demonstrating that \(S_{n,k}\) always has a uniform shortest path routing.

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References

  1. Araki, T.: Hyper Hamiltonian laceability of Cayley graphs generated by transpositions. Networks 48, 121–124 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cheng, E., Grossman, J., Qiu, K., Shen, Z.: Distance formula and shortest paths of the \((n, k)\)-star graphs. Inf. Sci. 180, 1671–1680 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cheng, E., Lipman, M.J.: Vulnerability issues of star graphs, alternating group graphs and split-stars: strength and toughness. Discret. Appl. Math. 118, 163–179 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cheng, E., Lipman, M.J.: Increasing the connectivity of the star graphs. Networks 40, 165–169 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cheng, E., Lipták, L., Shawash, N.: Orienting Cayley graphs generated by transposition trees. Comput. Math. Appl. 55, 2662–2672 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheng, E., Lipták, L., Steffy, D.E.: Strong local diagnosability of \((n, k)\)-star graphs and Cayley graphs generated by 2-trees with missing edges. Inf. Process. Lett. 113, 452–456 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheng, E., Qiu, K., Shen, Z.: A short note on the surface area of star graphs. Parallel Process. Lett. 19, 19–22 (2009)

    Article  MathSciNet  Google Scholar 

  8. Cheng, E., Qiu, K., Shen, Z.: A generating function approach to the surface area of some interconnection networks. J. Interconnect. Netw. 10, 189–204 (2009)

    Article  Google Scholar 

  9. Cheng, E., Qiu, K., Shen, Z.: A note on the alternating group network. J. Supercomput. 59, 246–248 (2012)

    Article  Google Scholar 

  10. Cheng, E., Qiu, K., Shen, Z.: The number of shortest paths in the \((n, k)\)-star graphs. Discret. Math. Algorithms Appl. 1450051 (2014). doi:10.1142/S1793830914500517

  11. Chiang, W.-K., Chen, R.-J.: The \((n, k)\)-star graph: a generalized star graph. Inf. Process. Lett. 56, 259–264 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chung Jr., F.R.K., Coffman, E.G., Reiman, M.I., Simon, B.: The forwarding index of communication networks. IEEE Trans. Inf. Theory 33, 224–232 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  13. Duh, D.-R., Chen, T.-L., Wang, Y.-L.: \((n-3)\)-edge-fault-tolerant weak-pancyclicity of \((n, k)\)-star graphs. Theor. Comput. Sci. 516, 28–39 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gargano, L., Vaccaro, U., Vozella, A.: Fault tolerant routing in the star and pancake interconnection networks. Inf. Process. Lett. 45, 315–320 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gauyacq, G.: Edge-forwarding index of star graphs and other Cayley graphs. Discr. Appl. Math. 80, 149–160 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gu, Q., Peng, S.: An efficient algorithm for \(k\)-pairwise disjoint paths in star graphs. Inf. Process. Lett. 67, 283–287 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gu, Q., Peng, S.: Cluster fault tolerant routing in star graphs. Networks 35, 83–90 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hungerford, T.: Algebra, Graduate Texts in Mathematics, vol. 73. Springer, New York (1980)

    Google Scholar 

  19. Heydemann, M.-C., Meyer, J.-C., Sotteau, D.: On the-forwarding index of networks. Discret. Appl. Math. 23, 103–123 (1989)

    Article  MATH  Google Scholar 

  20. Hoelzeman, D.A., Bettayeb, S.: On the genus of the star graphs. IEEE Trans. Comput. 43, 755–759 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hsu, H.-C., Hsieh, Y.-L., Tan, J.J.M., Hsu, L.H.: Fault Hamiltonicity and fault Hamiltonian connectivity of the \((n, k)\)-star graphs. Networks 42, 189–201 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Irving, J., Ratton, A.: Minimal factorizations of permutations into star transpositions. Discret. Math. 309, 1549–1554 (2009)

    Article  MathSciNet  Google Scholar 

  23. Ji, Y.: A new class of Cayley networks based on the alternating groups. Appl. Math. A J. Chin. Univ. 14, 235–239 (1999)

    MATH  Google Scholar 

  24. Kumar, V., Grama, A., Gupta, A., Karypis, G.: Introduction to Parallel Computing: Design and Analysis of Algorithms. Benjamin/Cummings, San Francisco (1994)

    MATH  Google Scholar 

  25. Latifi, S.: A study of fault tolerance in star graph. Inf. Process. Lett. 102, 196–200 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, T.-K., Tan, J.J.M., Hsu, L.-H.: Hyper hamiltonian laceability on edge fault star graph. Inf. Sci. 165, 59–71 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mendia, V.E., Sarkar, D.: Optimal broadcasting on the star graphs. IEEE Trans. Parallel Distrib. Syst. 3, 389–396 (1992)

    Article  MathSciNet  Google Scholar 

  28. Sabidussi, G.: On a class of fixed-point-free graphs. Proc. Am. Math. Soc. 5, 800–804 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sheu, J.P., Wu, C.Y., Chen, T.S.: An optimal broadcasting algorithm without message redundancy in star graphs. Trans. Parallel Distrib. Syst. 6, 653–658 (1995)

    Article  Google Scholar 

  30. Shen, X., Hu, Q., Cong, B., Sudborough, H., Girou, M., Bettayeb, S.: The 4-star graph is not a subgraph of any hypercube. Inf. Process. Lett. 37, 199–203 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  31. Shen, Z., Qiu, K., Cheng, E.: On the surface area of the \((n, k)\)-star graph. Theor. Comput. Sci. 410, 5481–5490 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Shim, S., Siran, J., Zerovnik, J.: Counterexamples to the uniform shortest path routing conjecture for vertex-transitive graphs. Discret. Appl. Math. 119, 281–286 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tanaka, Y., Kikuchi, Y., Araki, T., Shibata, Y.: Bipancyclic properties of Cayley graphs generated by transpositions. Discret. Math. 310, 748–754 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tseng, Y.C., Chen, Y.S., Juang, T.Y., Chang, C.J.: Congestion-free, dilation-2 embedding of complex binary trees into star graphs. Networks 33, 221–231 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tseng, Y.C., Sheu, J.P.: Toward optimal broadcast in a star graph using multiple spanning tree. IEEE Trans. Comput. 46, 593–599 (1997)

    Article  MathSciNet  Google Scholar 

  36. Wei, Y., Chen, F.: Generalized connectivity of \((n, k)\)-star graphs. Int. J. Found. Comput. Sci. 24, 1235–1241 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Xiang, Y., Stewart, I.A.: One-to-many node-disjoint paths in \((n, k)\)-star graphs. Discret. Appl. Math. 158, 62–70 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yeh, C.H., Parhami, B.: VLSI layouts of complete graphs and star graphs. Inf. Process. Lett. 68, 39–45 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  39. Yuan, A., Cheng, E., Lipták, L.: Linearly many faults in \((n, k)\)-star graphs. Int. J. Found. Comput. Sci. 22, 1729–1745 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhou, S.M., Xiao, W.J., Parhami, B.: Construction of vertex-disjoint paths in alternating group networks. J. Supercomput. 54, 206–228 (2010)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Oakland University, Rochester, MI, 48309, USA

    Eddie Cheng, Li Li, László Lipták & Daniel E. Steffy

  2. Department of Engineering, Robert Morris University, 6001 University Blvd., Moon Township, PA, 15108, USA

    Sangho Shim

Authors
  1. Eddie Cheng
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  2. Li Li
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  3. László Lipták
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  4. Sangho Shim
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  5. Daniel E. Steffy
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Correspondence to Daniel E. Steffy.

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Cheng, E., Li, L., Lipták, L. et al. On the Problem of Determining which (n, k)-Star Graphs are Cayley Graphs. Graphs and Combinatorics 33, 85–102 (2017). https://doi.org/10.1007/s00373-016-1741-8

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  • DOI: https://doi.org/10.1007/s00373-016-1741-8

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