There are no excess one digraphs

TL;DR

The paper addresses the directed degree/diameter problem by examining whether excess-one digraphs exist, i.e., $k$-geodetic digraphs with order $M(d,k)+1$. It develops a quotient construction, the orbital-connection digraph, based on the outlier automorphism and uses spectral constraints to bound the structure, reducing to a finite set of exceptional $(d,k)$. The authors then rule out all exceptional cases through detailed combinatorial and group-action arguments, establishing nonexistence for all $d,k\ge 2$. This result generalizes earlier nonexistence results for undirected graphs and informs the structure of near-Moore digraphs, including implications for almost Moore digraphs and higher excess.

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.

Figures (2)

• Figure 1: Geodetic cages for $d=2,k=3$ with excess $\epsilon = 5$
• Figure 2: The existence of two distinct walks of lengths $\leq k/2$ from $v(r)$ to $o^a(v(r))$ and $o^b(v(r))$ with $a \neq b$ would imply the existence of two distinct walks of length $\leq k$ from $v(r)$ to $o^{a+b}(v(r))$.

Theorems & Definitions (21)

• Lemma 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.6