Weak saturation of tensor product of cliques

TL;DR

This paper determines weak saturation numbers for tensor products of cliques and their colored variants, extending previous results and generalizing to arbitrary families of hypergraphs.

Contribution

It generalizes known weak saturation results to tensor products of cliques and their colored versions, including arbitrary families of hypergraphs.

Findings

Calculated weak saturation numbers for tensor product of cliques.

Extended results to colored weak saturation numbers for unions of tensor products.

Generalized to arbitrary families of hypergraphs for colored weak saturation.

Abstract

Given two hypergraphs $G$ and $H$, the weak saturation number $\operatorname{\mathrm{wsat}}(G,H)$ is the minimum number of edges in a spanning subhypergraph $F$ of $G$ such that the missing edges of $F$ can be added one at a time so that each added edge creates a copy of $H$. In this work, we determine weak saturation numbers for the case when $G$ and $H$ are tensor product of cliques, generalizing a result of Moshkovitz and Shapira (Journal of Combinatorial Theory, Series B, 2015), who found the exact values of $\operatorname{\mathrm{wsat}}(K^d_{n_1,\ldots,n_d},\ K^d_{r_1,\ldots,r_d})$. The proof also yields results for colored weak saturation numbers $\operatorname{\mathrm{c-wsat}}(G,H)$ of colored hypergraphs $G$ and $H$, where the colorings of the copies of $H$ must be compatible with the coloring of $G$. We determine these numbers when $G$ and $H$ are unions of tensor product of cliques, generalizing a result of Bulavka, Tancer, and Tyomkyn (Combinatorica, 2023), who determined $\operatorname{\mathrm{c-wsat}}(K^q_{n_1,\ldots,n_d}, K^q_{r_1,\ldots,r_d})$. Moreover, our proof allows us to generalize a result of Balogh, Bollob\'{a}s, Morris, and Riordan (Journal of Combinatorial Theory, Series A, 2012) by determining colored weak saturation numbers $\operatorname{\mathrm{c-wsat}}(K^d_{n_1,\ldots,n_d},\{K^d_{r_1,\ldots,r_d}\}_{\mathbf{r}\in \mathcal{R}})$ for an arbitrary family $\mathcal{R}$. The quantity $\operatorname{\mathrm{c-wsat}}(G,\mathcal{H})$ extends colored weak saturation by allowing, at each step, the creation of a colored copy of any hypergraph in the fixed family of hypergraphs $\mathcal{H}$.