A generalization of Turán's theorem to directed graphs (original) (raw)

1980, Discrete Mathematics

We consider An extremal problem for directed graphs which is closely related to Tutin's theorem giving the maximum number of edges in a gr;lph on n vertices which does not contain a complete subgraph on m vertices. For an ;ntc&r n 22, let T,, denote the transitive tournament with vertex set X,, = {1,2,3,. .. , n) and edge set {(i. j): 1 s i C j s n]. A subgraph H of T,, is said to be m-locally unipathic when the restriction of H to each m element subset of X,, consisting of m consecutive integers is unipathic. We show that the maximum number of edges in a m-locally unipathic subgraph of T,, is (;g)(m-l)'+q(m-1)r + Ur"] where n = q(m-1) + r and [$<m-I)] s rc' @rn-1)1. As is the case with T&n's theorem, the extremal graphs for our problem are complete multipartite graphs. Unlike T:r&n's theorem, the part sizes will not be uniform The proof of our principal theorem rests 011 a combiaatorial theory originally developed to inves:dgate the rank of partial'iy ordered sets. For integers, n, k with n se k a 2, let g(n, k) be the maximum number of edges in a graph G on n vertices which does not contain a complete subgraph on k vertices. Then let n = (k-1)q + r where 0 6 r C k-1 and consider the complete multipartite graph G(n, k) having k-1r parts of size q and r parts of size q f 1. Clearly, G(n, k) has n vertices but does not have a complete subgraph on k vertices. The following well known theorem of P'. Tur6n [9] tell us that the lower bound on g(n, k) provided by the graph G(n, k) is best possible. It also Sells us that G(n, k) is the unique extremal graph. Theorem 1 (Turhn). For integers m, k with n > k 2~ 2 the maximurn number g(n, k) of edges in a graph on n vertices which does not contain a complete subgraph on k