There are no excess one digraphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918228491501568 |
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| author | Filipovski, Slobodan Messegué, Arnau Miret, Josep M. Tuite, James |
| author_facet | Filipovski, Slobodan Messegué, Arnau Miret, Josep M. Tuite, James |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02827 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | There are no excess one digraphs Filipovski, Slobodan Messegué, Arnau Miret, Josep M. Tuite, James Combinatorics 05C12, 05C20, 05C35 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. |
| title | There are no excess one digraphs |
| topic | Combinatorics 05C12, 05C20, 05C35 |
| url | https://arxiv.org/abs/2512.02827 |