Upper bounds for linear graph codes
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866929324741885952 |
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| author | Versteegen, Leo |
| author_facet | Versteegen, Leo |
| contents | A linear graph code is a family $\mathcal{C}$ of graphs on $n$ vertices with the property that the symmetric difference of the edge sets of any two graphs in $\mathcal{C}$ is also the edge set of a graph in $\mathcal{C}$. In this article, we investigate the maximal size of a linear graph code that does not contain a copy of a fixed graph $H$. In particular, we show that if $H$ has an even number of edges, the size of the code is $O(2^{\binom{n}{2}}/\log n)$, making progress on a question of Alon. Furthermore, we show that for almost all graphs $H$ with an even number of edges, there exists $\varepsilon_H>0$ such that the size of a linear graph code without a copy of $H$ is at most $2^{\binom{n}{2}}/n^{\varepsilon_H}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_19891 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Upper bounds for linear graph codes Versteegen, Leo Combinatorics A linear graph code is a family $\mathcal{C}$ of graphs on $n$ vertices with the property that the symmetric difference of the edge sets of any two graphs in $\mathcal{C}$ is also the edge set of a graph in $\mathcal{C}$. In this article, we investigate the maximal size of a linear graph code that does not contain a copy of a fixed graph $H$. In particular, we show that if $H$ has an even number of edges, the size of the code is $O(2^{\binom{n}{2}}/\log n)$, making progress on a question of Alon. Furthermore, we show that for almost all graphs $H$ with an even number of edges, there exists $\varepsilon_H>0$ such that the size of a linear graph code without a copy of $H$ is at most $2^{\binom{n}{2}}/n^{\varepsilon_H}$. |
| title | Upper bounds for linear graph codes |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2310.19891 |