Quasi‐carousel tournaments.

Abstract: A tournament is called locally transitive if the outneighborhood and the inneighborhood of every vertex are transitive. Equivalently, a tournament is locally transitive if it avoids the tournaments <italic>W</italic>4 and <italic>L</italic>4, which are the only tournaments up to isomorphism on four vertices containing a unique 3‐cycle. On the other hand, a sequence of tournaments  ( T n ) n ∈ N with  V ( T n ) = n is called almost balanced if all but  o ( n ) vertices of  T n have outdegree  ( 1 / 2 + o ( 1 ) ) n. In the same spirit of quasi‐random properties, we present several characterizations of tournament sequences that are both almost balanced and asymptotically locally transitive in the sense that the density of <italic>W</italic>4 and <italic>L</italic>4 in  T n goes to zero as <italic>n</italic> goes to infinity. [ABSTRACT FROM AUTHOR]