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arXiv · 2512.02827

There are no excess one digraphs

Slobodan Filipovski·Arnau Messegué·Josep M. Miret·James Tuite
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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.

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Slobodan Filipovski, Arnau Messegué, Josep M. Miret, James Tuite. 2025-12-02. There are no excess one digraphs. https://arxiv.org/abs/2512.02827

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