A Note on Mixed Cages of Girth 5
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913750256189440 |
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| author | Araujo-Pardo, Gabriela Mendoza-Cadena, Lydia Mirabel |
| author_facet | Araujo-Pardo, Gabriela Mendoza-Cadena, Lydia Mirabel |
| contents | A mixed regular graph is a graph where every vertex has $z$ incoming arcs, $z$ outgoing arcs, and $r$ edges; furthermore, if it has girth $g$, we say that the graph is a \emph{$[z,r;g]$-mixed graph}. A \emph{$[z,r;g]$-mixed cage} is a $[z,r;g]$-mixed graph with the smallest possible order. In this note, we give a family of $[z,q;5]$-mixed graphs for $q\geq 7$ power of prime and $q-1\leq 4z+R$ with $z\geq 1$ and $R \in \{1,\ldots,5\}$. This provides better upper bounds on the order of mixed cages until this moment. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17152 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Note on Mixed Cages of Girth 5 Araujo-Pardo, Gabriela Mendoza-Cadena, Lydia Mirabel Combinatorics A mixed regular graph is a graph where every vertex has $z$ incoming arcs, $z$ outgoing arcs, and $r$ edges; furthermore, if it has girth $g$, we say that the graph is a \emph{$[z,r;g]$-mixed graph}. A \emph{$[z,r;g]$-mixed cage} is a $[z,r;g]$-mixed graph with the smallest possible order. In this note, we give a family of $[z,q;5]$-mixed graphs for $q\geq 7$ power of prime and $q-1\leq 4z+R$ with $z\geq 1$ and $R \in \{1,\ldots,5\}$. This provides better upper bounds on the order of mixed cages until this moment. |
| title | A Note on Mixed Cages of Girth 5 |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.17152 |