Isomorphy up to complementation

TL;DR

This paper establishes conditions under which two uniform hypergraphs are identical up to complementation based on partial isomorphisms, providing explicit formulas for key parameters and extending previous results.

Contribution

It proves the existence of specific parameters ensuring hypergraph equivalence up to complementation and determines exact values for these parameters for all uniformities.

Findings

Defined parameters s(h) and v(h) for hypergraph isomorphism up to complementation.

Derived explicit formula s(h)=h+2^{loor{\log_2 h floor}} for all h.

Established bounds for v(h) in special cases where h=2^{ ext{ extlangle}\ell angle} or h=2^{ ext{ extlangle}\ell angle}+1.

Abstract

Considering uniform hypergraphs, we prove that for every non-negative integer $h$ there exist two non-negative integers $k$ and $t$ with $k\leq t$ such that two $h$-uniform hypergraphs ${\mathcal H}$ and ${\mathcal H}'$ on the same set $V$ of vertices, with $| V| \geq t$, are equal up to complementation whenever ${\mathcal H}$ and ${\mathcal H}'$ are $k$-{hypomorphic up to complementation}. Let $s(h)$ be the least integer $k$ such that the conclusion above holds and let $v(h)$ be the least $t$ corresponding to $k=s(h)$. We prove that $s(h)= h+2^{\lfloor \log_2 h\rfloor} $. In the special case $h=2^{\ell}$ or $h=2^{\ell}+1$, we prove that $v(h)\leq s(h)+h$. The values $s(2)=4$ and $v(2)=6$ were obtained in a previous work.