Abstract
AbstractA digraph G is k-geodetic if for any pair of (not necessarily distinct) vertices $$u,v \in V(G)$$
u
,
v
∈
V
(
G
)
there is at most one walk of length $$\le k$$
≤
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 $$k = 2$$
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.
Funder
Engineering and Physical Sciences Research Council
London Mathematical Society
Nemzeti Kutatasi, Fejlesztesi es Innovacios Alap
Publisher
Springer Science and Business Media LLC
Subject
Discrete Mathematics and Combinatorics,Theoretical Computer Science
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