There are no excess one digraphs

TL;DR

This paper proves that no $k$-geodetic digraphs with minimum out-degree $d \,\geq 2$ and order exactly one more than the Moore bound exist, confirming a conjecture and extending non-existence results.

Contribution

It proves the conjecture that excess one $k$-geodetic digraphs do not exist for $d,k \,\geq 2$, complementing known results for undirected graphs.

Findings

No excess one $k$-geodetic digraphs for $d,k \,\geq 2$

Supports the non-existence conjecture for these digraphs

Extends non-existence results from undirected to directed graphs

Abstract

A digraph $G$ is \emph{$k$-geodetic} if for any pair $u,v \in V(G)$ there is at most one $u,v$-walk of length not exceeding $k$. The order of a $k$-geodetic digraph with minimum out-degree $d$ is bounded below by the directed Moore bound $M(d,k) = 1 + d + d^2+ \cdots +d^k$. It is known that the Moore bound cannot be achieved for $d,k \geq 2$. A $k$-geodetic digraph with minimum degree $d$ and order one greater than the Moore bound has \emph{excess one}. In this paper we prove a conjecture that no excess one digraphs exist for $d,k \geq 2$, thus complementing the result of Bannai and Ito on the non-existence of undirected graphs with excess one.