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Trail-connected tournaments

Author

Listed:
  • Liu, Juan
  • Liu, Qinghai
  • Zhang, Xindong
  • Chen, Xing

Abstract

If for any two vertices v1 and v2 of digraph D, D admits a spanning (v1, v2)-dipath or a spanning (v2, v1)-dipath, then D is weakly Hamiltonian-connected; and if there are both a spanning (v1, v2)-dipath and a spanning (v2, v1)-dipath, then D is strongly Hamiltonian-connected. Thomassen in [J. of Combinatorial Theory, Series B, 28, (1980) 142–163] discovered a collection T0 of two digraphs and used it to characterize weakly Hamiltonian-connected tournaments, and proved that every 4-connected tournament is strongly Hamiltonian-connected. For any two vertices v1 and v2 of digraph D, D is weakly trail-connected if D admits a spanning (v1, v2)-ditrail or a spanning (v2, v1)-ditrail, and D is strongly trail-connected if there are both a spanning (v1, v2)-ditrail and a spanning (v2, v1)-ditrail. We have determined a family T of tournaments and prove the following. (i) A tournament D is weakly trail-connected if and only if D is strong. (ii) A strong tournament D is strongly trail-connected if and only if D is not a member in T. (iii) Every tournament with arc-strong connectivity at least 2 is strongly trail-connected.

Suggested Citation

  • Liu, Juan & Liu, Qinghai & Zhang, Xindong & Chen, Xing, 2021. "Trail-connected tournaments," Applied Mathematics and Computation, Elsevier, vol. 389(C).
    • Handle: RePEc:eee:apmaco:v:389:y:2021:i:c:s0096300320305506
    DOI: 10.1016/j.amc.2020.125595
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