Bounds and constructions for n-e.c. tournaments

Few families of tournaments satisfying the $n$-e.c. adjacency property are known. We supply a new random construction for generating infinite families of vertex-transitive $n$-e.c. tournaments by considering circulant tournaments. Switching is used to generate exponentially many $n$-e.c. tournaments of certain orders. With aid of a computer search, we demonstrate that there is a unique minimum order $3$-e.c. tournament of order $19,$ and there are no $3$-e.c. tournaments of orders $20,$ $21,$ and $22.$