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On cubic 2-independent Hamiltonian connected graphs

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Abstract

A graph G=(V,E) is Hamiltonian connected if there exists a Hamiltonian path between any two vertices in G. Let P 1=(u 1,u 2,…,u |V|) and P 2=(v 1,v 2,…,v |V|) be any two Hamiltonian paths of G. We say that P 1 and P 2 are independent if u 1=v 1,u |V|=v |V|, and u i ≠v i for 1<i<|V|. A cubic graph G is 2-independent Hamiltonian connected if there are two independent Hamiltonian paths between any two different vertices of G. A graph G is 1-Hamiltonian if G−F is Hamiltonian for any F⊆V∪E with |F|=1. A graph G is super 3*-connected if there exist i internal disjoint paths spanning G for i=1,2,3. It is proved that every super 3*-connected graph is 1-Hamiltonian. In this paper, we prove that every cubic 2-independent Hamiltonian connected graph is also 1-Hamiltonian. We present some cubic 2-independent Hamiltonian connected graphs that are super 3*-connected, some cubic 2-independent Hamiltonian connected graphs that are not super 3*-connected, some super 3*-connected graphs that are not 2-independent Hamiltonian connected, and some cubic 1-Hamiltonian graphs that are Hamiltonian connected but neither 2-independent Hamiltonian connected nor super 3*-connected.

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References

  • Albert M, Aldred ERI, Holton D, Sheehan J (2001) On globally 3*-connected graphs. Australas J Comb 24:193–207

    MATH  Google Scholar 

  • Bondy JA, Murty USR (1980) Graph theory with applications. North-Holland, New York

    Google Scholar 

  • Grinberg EJ (1968) Plane homogeneous graphs of degree three without Hamiltonian circuits. Latv Math Yearb 5:51–58

    Google Scholar 

  • Harary F, Hayes JP (1993) Edge fault tolerance in graphs. Networks 23:135–142

    Article  MATH  Google Scholar 

  • Harary F, Hayes JP (1996) Node fault tolerance in graphs. Networks 27:19–23

    Article  MATH  Google Scholar 

  • Kao SS, Huang HM, Hsu NM, Hsu LH (2007) Cubic 1-Hamiltonian graphs, globally 3*-connected graphs, and super 3*-connected graphs. Discret Appl Math (submitted)

  • Lin CK, Huang HM, Hsu LH (2005) The super connectivity of the pancake graphs and the super laceability of the star graphs. Theor Comput Sci 39:257–271

    Article  Google Scholar 

  • Lin CK, Huang HM, Hsu LH, Bau S (2005) Mutually Independent Hamiltonian paths in star networks. Networks 46:100–117

    Article  Google Scholar 

  • Moon JW (1965) On a problem of Ore. Math Gaz 49:40–41

    Article  MATH  Google Scholar 

  • Ore O (1963) Hamiltonian connected graphs. J Math Pure Appl 42:21–27

    MATH  Google Scholar 

  • Wang JJ, Hung CN, Tan JM, Hsu LH, Sung TY (2000) Construction schemes for fault tolerant Hamiltonian graphs. Networks 35:233–245

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Industrial Engineering and Management, Ta Hwa Institute of Technology, Hsinchu County, 307, Taiwan

    Tung-Yang Ho

  2. Department of Computer Science and Information Engineering, Da-Yeh University, Changhua County, 515, Taiwan

    Chun-Nan Hung

  3. Department of Computer Science and Information Engineering, Providence University, Taichung, 433, Taiwan

    Lih-Hsing Hsu

Authors
  1. Tung-Yang Ho
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  2. Chun-Nan Hung
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  3. Lih-Hsing Hsu
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Corresponding author

Correspondence to Tung-Yang Ho.

Additional information

Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday.

This work was supported in part by the National Science Council of the Republic of China under Contract NSC 94-2213-E-233-011.

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Ho, TY., Hung, CN. & Hsu, LH. On cubic 2-independent Hamiltonian connected graphs. J Comb Optim 14, 275–294 (2007). https://doi.org/10.1007/s10878-007-9054-8

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  • DOI: https://doi.org/10.1007/s10878-007-9054-8

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