Transversal Structures in Graph Systems: A Survey

TL;DR

This survey reviews extremal problems related to transversal structures in systems of graphs, digraphs, and hypergraphs, highlighting conditions for their existence and discussing open problems.

Contribution

It summarizes sufficient conditions for transversal structures in graph systems, extending classical extremal graph theory results to the transversal setting.

Findings

Summarizes conditions ensuring transversal structures exist

Generalizes classical extremal theorems to transversal versions

Includes conjectures and open problems in the field

Abstract

Given a system $\mathcal{G} =\{G_1,G_2,\dots,G_m\}$ of graphs/digraphs/hypergraphs on the common vertex set $V$ of size $n$, an $m$-edge graph/digraph/hypergraph $H$ on $V$ is transversal in $\mathcal{G}$ if there exists a bijection $\phi:E(H)\rightarrow [m]$ such that $e \in E(G_{\phi(e)})$ for all $e\in E(H)$. In this survey, we consider extremal problems for transversal structures in graph systems. More precisely, we summarize some sufficient conditions that ensure the existence of transversal structures in graph/digraph/hypergraph systems, which generalize several classical theorems in extremal graph theory to transversal version. We also include a number of conjectures and open problems.