Difference-Isomorphic Graph Families
arxiv(2023)
摘要
Many well-studied problems in extremal combinatorics deal with the maximum
possible size of a family of objects in which every pair of objects satisfies a
given restriction. One problem of this type was recently raised by Alon,
Gujgiczer, K\"orner, Milojevi\'c and Simonyi. They asked to determine the
maximum size of a family $\mathcal{G}$ of graphs on $[n]$, such that for every
two $G_1,G_2 \in \mathcal{G}$, the graphs $G_1 \setminus G_2$ and $G_2
\setminus G_1$ are isomorphic. We completely resolve this problem by showing
that this maximum is exactly $2^{\frac{1}{2}\big(\binom{n}{2} - \lfloor
\frac{n}{2}\rfloor\big)}$ and characterizing all the extremal constructions. We
also prove an analogous result for $r$-uniform hypergraphs.
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