Realization of certain generalized paths in tournaments

A tournament T,, consists of a set of n vertices and a single directed edge joining every pair of distinct vertices. We denote the vertices of T’. by {l, . . . , n). A permutation aI, . . . , a, of the vertices is a generalized path. The type af the path is characterized by the sequence a,_1 = e3 l l l E~__~ where si = i-1 if e -+ ai+1 and 6 = -1 if a, -+ ai tl. We also say that mn_l is realized i: T.. In [S], it was conjectured that if it g? 8, then every tournament T, realizes all possible 2*-l types u,._~. Certain types are known to be always realizable. Thus, since every tournament has a Hamiltonian path the types & = +1, -t1, . . . r +1 and (or= 1, -l,..., -1 are always realizable. Gaiinbaum (Harary [3, p. 21 I, ex. 16.261) observed that if n 2 b, then every Tfl has a Hamiltonian path al + 9 l + a, with a, tal. Thus, types irt which all 6 but one have the same sign are realizable. Griinbaum [2] and Rosenfeld [4] proved that every tournament with three exceptions, has an attidirected Hamiltonian path. The exceptions are the regular tournaments TR,, TRS, TR,, when= TR, is the only regulw tournament on 7 vcr’ices with no transitive sub-tournament on 4 vertices. Thus every T,, n a 8 realizes the type o,.__~ = e1 l l l E,_~~ &i = fl)i. Finally, Forcade [l] proved that if n = 2k then every type o,__~ is realizable by T,,. The purpose of this note is to prove that sequences with “larg; blocks” are always realizable.