Trail-connected tournaments

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.