Extremal Problems For Transversals In Graphs With Bounded Degree

We introduce and discuss generalizations of the problem of independent transversals. Given a graph property ${\user1{\mathcal{R}}}$, we investigate whether any graph of maximum degree at most d with a vertex partition into classes of size at least p admits a transversal having property ${\user1{\mathcal{R}}}$. In this paper we study this problem for the following properties ${\user1{\mathcal{R}}}$: “acyclic”, “H-free”, and “having connected components of order at most r”.

We strengthen a result of [13]. We prove that if the vertex set of a d-regular graph is partitioned into classes of size d+⌞d/r⌟, then it is possible to select a transversal inducing vertex disjoint trees on at most r vertices. Our approach applies appropriate triangulations of the simplex and Sperner’s Lemma. We also establish some limitations on the power of this topological method.

We give constructions of vertex-partitioned graphs admitting no independent transversals that partially settles an old question of Bollobás, Erdős and Szemerédi. An extension of this construction provides vertex-partitioned graphs with small degree such that every transversal contains a fixed graph H as a subgraph.

Finally, we pose several open questions.