A general approach to transversal versions of Dirac‐type theorems

Abstract

AbstractGiven a collection of hypergraphs with the same vertex set, an ‐edge graph is a transversal if there is a bijection such that for each . How large does the minimum degree of each need to be so that necessarily contains a copy of that is a transversal? Each in the collection could be the same hypergraph, hence the minimum degree of each needs to be large enough to ensure that . Since its general introduction by Joos and Kim (Bull. Lond. Math. Soc. 52 (2020) 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 graphs on an ‐vertex set, each with minimum degree at least , contains a transversal copy of the th power of a Hamilton cycle. This can be viewed as a rainbow version of the Pósa–Seymour conjecture.