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arXiv · 2301.13305

Graph-codes

Noga Alon
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Abstract

The symmetric difference of two graphs $G_1,G_2$ on the same set of vertices $[n]=\{1,2, \ldots ,n\}$ is the graph on $[n]$ whose set of edges are all edges that belong to exactly one of the two graphs $G_1,G_2$. Let $H$ be a fixed graph with an even (positive) number of edges, and let $D_H(n)$ denote the maximum possible cardinality of a family of graphs on $[n]$ containing no two members whose symmetric difference is a copy of $H$. Is it true that $D_H(n)=o(2^{n \choose 2})$ for any such $H$? We discuss this problem, compute the value of $D_H(n)$ up to a constant factor for stars and matchings, and discuss several variants of the problem including ones that have been considered in earlier work.

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Noga Alon. 2023-02-06. Graph-codes. https://arxiv.org/abs/2301.13305

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