Partite Saturation Problems

TL;DR

This paper investigates minimal edge counts in complete blow-ups of graphs to prevent certain partite subgraphs, providing exact saturation numbers for various graphs including $K_4$, paths, and stars, with results depending on graph connectivity.

Contribution

It extends saturation problem analysis to complete blow-ups of graphs, calculating new saturation numbers for $K_4$ and other graph classes, and explores related variants.

Findings

Calculated saturation number for $K_4$ in large $n$ case.

Provided exact saturation numbers for paths and stars.

Identified linear and quadratic growth of saturation numbers based on graph connectivity.

Abstract

We look at several saturation problems in complete balanced blow-ups of graphs. We let $H[n]$ denote the blow-up of $H$ onto parts of size $n$ and refer to a copy of $H$ in $H[n]$ as 'partite' if it has one vertex in each part of $H[n]$. We then ask how few edges a subgraph $G$ of $H[n]$ can have such that $G$ has no partite copy of $H$ but such that the addition of any new edge from $H[n]$ creates a partite $H$. When $H$ is a triangle this value was determined by Ferrara, Jacobson, Pfender, and Wenger. Our main result is to calculate this value for $H=K_4$ when $n$ is large. We also give exact results for paths and stars and show that for $2$-connected graphs the answer is linear in $n$ whilst for graphs which are not $2$-connected the answer is quadratic in $n$. We also investigate a similar problem where $G$ is permitted to contain partite copies of $H$ but we require that the addition of any new edge from $H[n]$ creates an extra partite copy of $H$. This problem turns out to be much simpler and we attain exact answers for all cliques and trees.