On density conditions for transversal trees in multipartite graphs

TL;DR

This paper explores the minimum density conditions needed in multipartite graphs to ensure the existence of specific spanning trees, such as transversals and Hamiltonian transversals, advancing extremal graph theory.

Contribution

It introduces the study of density thresholds for transversal trees in multipartite graphs, providing new results, constructions, and conjectures in this area.

Findings

Derived bounds for density to guarantee transversal trees

Constructed examples illustrating extremal cases

Proposed conjectures and open questions for future research

Abstract

Let $G$ be an $r$-partite graph such that the edge density between any two parts is at least $\alpha$. How large does $\alpha$ need to be to guarantee that $G$ contains a connected transversal, that is, a tree on $r$ vertices meeting each part in one vertex? And what if instead we want to guarantee the existence of a Hamiltonian transversal? In this paper we initiate the study of such extremal multipartite graph problems, obtaining a number of results and providing many new constructions, conjectures and further questions.