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

Extremal digraphs avoiding distinct walks of length 4 with the same endpoints

“…In 2011, Huang, Zhan [5] solved the case k ≥ n − 1 and determined the extremal numbers for the cases k = n − 2 and k = n − 3. In 2019, Huang, Lyu and Qiao [4] characterized the extremal digraphs for the cases k = n − 2 and k = n − 3. And they gave the solutions to the case 5 ≤ k ≤ n − 4.…”

“…Remark that the cases n ∈ {5, 6, 7} have been solved in [4,5]. In this paper, we also characterize the extremal digraphs for n ∈ {8, 9, 10, 11}.…”

“…The Turán-type problem is one of the hottest topics in extremal graph theory, which concerns the number of edges in graphs containing no given subgraphs and the extremal graphs achieving this maximum. Most of the previous results of Turán problems concern undirected graphs and only a few Turán problems on digraphs have been investigated; see [1][2][3][4][5][6].…”

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

“…The initial version of Problem 1 was posed by Zhan at a seminar in 2007, which concerned the case t = 1, see [13, p. 234]. In the last decade, Problem 1 for the case t = 1 has been completely solved by Wu [12], by Huang and Zhan [8], by Huang, Lyu and Qiao [7], by Lyu [11], and by Huang and Lyu [6]. For the general cases of Problem 1, the case k = 2 has been studied in [9], and the case for k ≥ n − 1 ≥ 6t + 1 has been solved in [5].…”

0-1 matrices with zero trace whose squares are 0-1 matrices

“…For strict digraphs, the solution to Problem 3 for the case k ≥ 5 follows straightforward from [5,6], since the extremal digraphs in [5,6] are loopless. In this paper we consider the case k = 2 for Problem 3 on strict digraphs.…”