There are no excess one digraphs

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 $d,k\ge 2$. 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 $d,k\ge 2$, thus complementing the result of Bannai and Ito on the non-existence of undirected graphs with excess one.