A general approach to transversal versions of Dirac-type theorems

TL;DR

This paper develops a unified framework to determine minimum degree conditions in hypergraph collections that guarantee the existence of transversal copies of certain subgraphs, extending classical Dirac-type theorems to a rainbow setting.

Contribution

It provides a general sufficient condition for asymptotic tightness of minimum degree bounds in transversal hypergraph problems, unifying and extending previous results.

Findings

Established a broad sufficient condition for transversal Dirac-type theorems.

Derived a rainbow version of the Pósa-Seymour conjecture for Hamilton cycle powers.

Unified approach recovers many known results and introduces new transversal variants.

Abstract

Given a collection of hypergraphs $\textbf{H}=(H_1,\ldots,H_m)$ with the same vertex set, an $m$-edge graph $F\subset \cup_{i\in [m]}H_i$ is a transversal if there is a bijection $\phi:E(F)\to [m]$ such that $e\in E(H_{\phi(e)})$ for each $e\in E(F)$. How large does the minimum degree of each $H_i$ need to be so that $\textbf{H}$ necessarily contains a copy of $F$ that is a transversal? Each $H_i$ in the collection could be the same hypergraph, hence the minimum degree of each $H_i$ needs to be large enough to ensure that $F\subseteq H_i$. Since its general introduction by Joos and Kim [Bull. Lond. Math. Soc., 2020, 52(3):498-504], a growing body of work has shown that in many cases this lower bound is tight. In this paper, we give a unified approach to this problem by providing a widely applicable sufficient condition for this lower bound to be asymptotically tight. This is general enough to recover many previous results in the area and obtain novel transversal variants of several classical Dirac-type results for (powers of) Hamilton cycles. For example, we derive that any collection of $rn$ graphs on an $n$-vertex set, each with minimum degree at least $(r/(r+1)+o(1))n$, contains a transversal copy of the $r$-th power of a Hamilton cycle. This can be viewed as a rainbow version of the P\'osa-Seymour conjecture.