# Connected-Intersecting Families of Graphs

## TL;DR

This paper investigates the maximum size of graph families where intersections satisfy a property, specifically fully characterizing the connected-intersecting case using linear algebraic methods.

## Contribution

It provides a complete solution to the maximum size problem for connected-intersecting graph families and introduces new bounds related to fixed subgraph unions.

## Key findings

- Maximum size is achieved by graphs containing a fixed spanning tree.
- Complete resolution of the connected-intersecting case.
- New lower bounds for unions of fixed subgraphs.

## Abstract

For a graph property $\mathcal{P}$ and a common vertex set $V = \{1, 2, \ldots, n\}$, a family of graphs on $V$ is \emph{$\mathcal{P}$-intersecting} iff $G \cap H$ satisfies $\mathcal{P}$ for all $G,H$ in the family. Addressing a question of Chung, Graham, Frankl, and Shearer, we explore---for various $\mathcal{P}$---the maximum cardinality among all $\mathcal{P}$-intersecting families of graphs. In the connected-intersecting case, we resolve the question completely by a short linear algebraic proof showing this maximum is attained by taking all graphs containing a fixed spanning tree (though we show other extremal constructions as well). We also present a new lower bound for containing unions of a fixed subgraph.