Digraphs that have at most one walk of a given length with the same endpoints

Abstract

a b s t r a c tLet Θ(n, k) be the set of digraphs of order n that have at most one walk of length k with the same endpoints. Let θ (n, k) be the maximum number of arcs of a digraph in Θ(n, k). We prove that if n ≥ 5 and k ≥ n − 1 then θ (n, k) = n(n − 1)/2 and this maximum number is attained at D if and only if D is a transitive tournament. θ (n, n − 2) and θ (n, n − 3) are also determined.

A Turán problem on digraphs avoiding distinct walks of a given length with the same endpoints

“…Without loss of generality, we assume a n−1,n−2 = a n−2,n−1 = 1. By Lemma 1 (ii) of [8], we obtain a n−1,n−1 = a n−2,n−2 = 0 and f (x 1 ) + f (y 1 ) = n − 5 ≥ 4.…”

“…Let w 5 = x and u 5 = y T . By Lemma 9 of [8], E is permutation similar to T 5 and A is permutation similar to…”

Digraphs that contain at most t distinct walks of a given length with the same endpoints

“…We define z(t) as the smallest integer such that if k ≥ n − 1 ≥ z(t), then D ∈ EX(n, F k,t+1 ) if and only if D is a transitive tournament. Huang and Zhan [8] proved that z(1) = 4. It follows from Theorem 2 that z(t) is well defined for each positive integer t and…”

“…Based on this fact, using induction on n, Lyu [10] obtained the following result. From [8] we get z(1) = 4. Hence, Conjecture 4 holds confirmly when t = 1.…”

Extremal digraphs avoiding an orientation of C4

“…Maurer, Rabinovitch and Trotter [18] studied the extremal transitive − → C 2 -free digraphs which contain at most one directed path from x to y for any two distinct vertices x, y. In [15,16,25], the authors studied the extremal digraphs which have no distinct walks of a given length k with the same initial vertex and the same terminal vertex.…”