Turán Problems for k -Geodetic Digraphs

Tuite, James; Erskine, Grahame and Salia, Nika (2023). Turán Problems for k -Geodetic Digraphs. Graphs and Combinatorics, 39(2), article no. 25.

DOI: https://doi.org/10.1007/s00373-023-02619-x

Abstract

A digraph G is k-geodetic if for any pair of (not necessarily distinct) vertices u, v∈V(G) there is at most one walk of length ≤k from u to v in G. In this paper, we determine the largest possible size of a k-geodetic digraph with a given order. We then consider the more difficult problem of the largest size of a strongly-connected k-geodetic digraph with a given order, solving this problem for k=2 and giving a construction which we conjecture to be extremal for larger k. We close with some results on generalised Turán problems for the number of directed cycles and paths in k-geodetic digraphs.

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