Year: 2019
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Berger, Aaron, Berkowitz, Ross, Devlin, Pat, Doppelt, Michael, Durham, Sonali, Murthy, Tessa, Vemuri, Harish
Institutions: Massachusetts Institute of Technology, Yale University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Yale University
OpenAlex ID: W2908459059