On k-geodetic digraphs with excess one

Abstract

A k-geodetic digraph G is a digraph in which, for every pair of vertices u and v (not necessarily distinct), there is at most one walk of length ≤ k from u to v. If the diameter of G is k, we say that G is strongly geodetic. Let N (d, k) be the smallest possible order for a k-geodetic digraph of In this paper, we will prove that a (d, k, 1)-digraph is always out-regular and that if it is not in-regular, then it must have 2 vertices of in-degree less than d, d vertices of in-degree d + 1 and the remaining vertices will have in-degree d. Furthermore, we will prove there exist no (2, 2, 1)-digraphs and no diregular (2, k, 1)-digraphs for k ≥ 3.

Digraphs with degree two and excess two are diregular

“…A k-geodetic digraph with minimum out-degree ≥ d and order M (d, k)+ǫ is said to be a (d, k, +ǫ)-digraph or to have excess ǫ. It was shown in [6] that there are no diregular (2, k, +1)-digraphs for k ≥ 2. In 2016 it was shown in [5] that digraphs with excess one must be diregular and that there are no (d, k, +1)-digraphs for k = 2, 3, 4 and sufficiently large d. In a separate paper [7], the present author has shown that (2, k, +2)-digraphs must be diregular with degree d = 2 for k ≥ 2.…”

“…Without loss of generality, v = u 3 . Hence Case A second (2, 2, +2)-digraph V (G) = {u, u 1 , u 2 , v, u 4 , u 5 , u6 , v 1 , v 4 } and O(v) = {u 1 , u 4 }. We have the configuration shown inFigure 4.…”

The structure of digraphs with excess one

“…A directed graph G is k-geodetic if for any pair of (not necessarily distinct) vertices u, v of G there is at most one directed walk from u to v in G with length ≤ k. The following analogue for directed graphs of the degree/girth problem was raised by Sillasen in [23]:…”

On diregular digraphs with degree two and excess three

“…of vertices that cannot be reached by ≤ k-paths from u; any element of this set is an outlier of u. It is known that (d, k, +1)-digraphs are out-regular with degree d [26]. For digraph G with excess ǫ = 1, the setvalued function O can be construed as an outlier function o, where for each vertex u of G the outlier o(u) of u is the unique vertex of G with d(u, o(u)) ≥ k + 1.…”