Extremal problems for transversals in graphs with bounded degree

We introduce and discuss generalizations of the problem of independent transversals. Given a graph property 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 R.. In this paper we study this problem for the following properties 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 Bollobas, Erdos and Szemeredi. 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.