Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Mutually independent Hamiltonianicity of Cartesian product graphs

  • 113 Accesses

Abstract

Two Hamiltonian cycles \(C_1=\langle u_0,u_1,u_2,...,u_{n-1},u_0 \rangle \) and \(C_2=\langle v_0,v_1,v_2,...,v_{n-1},v_0 \rangle \) of a graph G are independent starting at \(u_0\) if \(u_0=v_0, u_i\ne v_i\) for all \(1\le i\le n-1\). A set of Hamiltonian cycles C of G are k-mutually independent starting at vertex u if any two different Hamiltonian cycles of C are independent starting at u and \(|C| = k\). The mutually independent Hamiltonianicity of graph G is the maximum integer k, such that for any vertex u of G there exist k-mutually independent Hamiltonian cycles starting at u, denoted by IHC(\(G)=k\). The Cartesian product of graphs G and H, written by \(G \times H\), is the graph with vertex set \(V(G) \times V(H)\) specified by putting (uv) adjacent to \((u', v')\) if and only if \((1)\;u = u'\) and \(vv' \in E(H),\) or \((2)\;v = v'\) and \(uu' \in E(G)\). In this paper, for \(G = G_1 \times G_2\), where \(G_1\) and \(G_2\) are Hamiltonian graphs, IHC(\(G_1 \times G_2) \ge \) IHC(\(G_1)\) or IHC(\(G_1)\) + 2 is proved when given some different conditions.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

References

  1. 1.

    Arabnia HR (1990) A parallel algorithm for the arbitrary rotation of digitized images using process-and-data-decomposition approach. J Parallel Distrib Comput 10(2):188–193

  2. 2.

    Arabnia HR (1995) A distributed stere correlation algorithm. Proceedings of Computer Communications and Networks (ICCCN’95), IEEE, 479–482

  3. 3.

    Arabnia HR, Bhandarkar SM (1996) Parallel stereocorrelation on a reconfigurable multi-ring network. J Supercomput 10(3):243–270

  4. 4.

    Arabnia HR, Smith JW (1993) A reconfigurable interconnection network for imaging operations and its implementation using a multi-stage switching box. Proceedings of the 7th Annual International High Performance Computing Conference. The 1993 High Performance Computing, New Horizons Supercomputing Symposium, Calgary, Alberta, Canada, 349–357

  5. 5.

    Arabnia HR, Oliver MA (1987) Arbitrary rotation of raster images with SIMD machine architectures. Comput Graph Forum 6(1):3–12

  6. 6.

    Arabnia HR, Oliver MA (1987) A transputer network for the arbitrary rotation of digitised images. Comput J 30(5):425–433

  7. 7.

    Arabnia HR, Oliver MA (1989) A transputer network for fast operations on digitised images. Comput Graph Forum 8(1):3–12

  8. 8.

    Bhandarkar SM, Arabnia HR (1995) The Hough transform on a reconfigurable multi-ring network. J Parallel Distrib Comput 24(1):107–114

  9. 9.

    Bhandarkar SM, Arabnia HR, Smith JW (1995) A reconfigurable architecture for image processing and computer vision. Int J Pattern Recognit Artif Intell 9(2):201–229

  10. 10.

    Bhandarkar SM, Arabnia HR (1995) The REFINE multiprocessor: theoretical properties and algorithms. Parallel Comput 21(11):1783–1806

  11. 11.

    Chang SYP, Juan JST, Lin CK, Tan JJM, Hsu LH (2009) Mutually independent hamiltonian connectivity of (\(n, k\))-star graphs. Ann Comb 13(1):27–52

  12. 12.

    Chartrand G, Oellermann OR (1993) Applied and algorithmic graph theory, vol 993. McGraw-Hill, New York

  13. 13.

    Hsieh SY, Lin TJ (2009) Panconnectivity and edge-pancyclicity of \(k\)-Ary \(n\)-cubes. Networks 54(1):1–11

  14. 14.

    Hsieh SY, Weng YF (2009) Fault-tolerant embedding of pairwise independent hamiltonian paths on a faulty hypercube with edge faults. Theory Comput Syst 45(2):407–425

  15. 15.

    Hsieh SY, Yu PY (2007) Fault-free mutually independent hamiltonian cycles in hypercubes with faulty edges. J Comb Optim 13(2):153–162

  16. 16.

    Kao SS, Pi-Hsiang (2010) Mutually independent hamiltonian cycles in \(k\)-ary \(n\)-cubes when k is odd. In: Proceedings of the 2010 American Conference on Applied Mathematics (AMERICAN- MATH 10). University of Harvard, Cambridge, Cambridge, United States, pp 116–121

  17. 17.

    Kueng TL, Liang T, Hsu LH (2008) Mutually independent hamiltonian cycles of binary wrapped buttery graphs. Math Comput Model 48(11–12):1814–1825

  18. 18.

    Kung TL, Lin CK, Liang T, Tan JJM, Hsu LH (2011) Fault-free mutually independent hamiltonian cycles of faulty star graphs. Int J Comput Math 88(4):731–746

  19. 19.

    Lai YL, Yu DC, Hsu LH (2011) Mutually independent hamiltonian cycle of burnt pancake graphs. IEICE Trans Fundam Electron Commun Comput Sci E94–A(7):1553–1557

  20. 20.

    Leighton FT (1992) Introduction to parallel algorithms and architectures. Morgan Kaufmann, San Francisco

  21. 21.

    Shih YK, Chuang HC, Kao SS, Tan JJM (2010) Mutually independent hamiltonian cycles in dual-cubes. J Supercomput 54(2):239–251

  22. 22.

    Su H, Chen SY, Kao SS (2012) Mutually independent hamiltonian cycles in alternating group graphs. J Supercomput 61(3):560–571

  23. 23.

    Su H, Pan JL, Kao SS (2011) Mutually independent hamiltonian cycles in \(k\)-ary \(n\)-cubes when \(k\) is even. Comput Electr Eng 37(3):319–331

  24. 24.

    Sun CM, Lin CK, Huang HM, Hsu LH (2006) Mutually independent hamiltonian paths and cycles in hypercubes. J Interconnect Netw 7(02):235–255

  25. 25.

    Tang KW, Padubidri SA (1994) Diagonal and toroidal mesh networks. IEEE Trans Comput 43(7):815–826

  26. 26.

    Teng YH, Tan JJ, Ho TY, Hsu LH (2006) On mutually independent hamiltonian paths. Appl Math Lett 19(4):345–350

  27. 27.

    Wang HR (2011) The balanced Hamiltonian cycle problem. Master’s thesis of Department of Computer Science and Information Engineering, National Chi Nan University

  28. 28.

    Wani MA, Arabnia HR (2003) Parallel edge-region-based segmentation algorithm targeted at reconfigurable multi-ring network. J Supercomput 25(1):43–63

  29. 29.

    Wu KS, Juan JST (2012) Mutually independent hamiltonian cycles of \(C_m \times C_n\) when \(m\), \(n\) are odd. In: Proceedings of the 29th Workshop on Combinatorial Mathematics and Computational Theory. National Taipei College of Business, Taipei, Taiwan, 165–170

  30. 30.

    Wu KS, Juan JST (2012) Mutually independent hamiltonian cycles of \(C_n \times C_n\). In: Proceeding of World Academy of Science, Engineering and Technology. Vol. 65. Tokyo, Japan, 754–760

  31. 31.

    Wu KS, Wang YC, Juan JST (2011) Mutually independent hamiltonian cycle of \(C_{2m+1} \times C_{2m+1}\). Proceedings of the 28th Workshop on Combinatorial Mathematics and Computational Theory, Penghu, Taiwan, 327–332

  32. 32.

    Xu J (2013) Topological structure and analysis of interconnection networks. Springer, US

  33. 33.

    Xu JM, Ma M (2009) Survey on path and cycle embedding in some networks. Front Math China 4(2):217–252

Download references

Acknowledgments

The authors would like to thank the anonymous reviewers for their helpful comments. This research was supported in part by the Ministry of Science and Technology, Taiwan, under Grant MOST 104-2221-E-260-005.

Author information

Correspondence to Justie Su-Tzu Juan.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wu, K., Wang, Y. & Juan, J.S. Mutually independent Hamiltonianicity of Cartesian product graphs. J Supercomput 73, 837–865 (2017). https://doi.org/10.1007/s11227-016-1804-x

Download citation

Keywords

  • Hamiltonian
  • Mutually independent
  • Hamiltonianicity
  • Cartesian product