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 Bollob

• Type: Journal
• Year: 2006
• Where: COMBINATORICA
• Authors: Tibor Szabó, Gábor Tardos
• Keywords: COMBINATORICA 2006, Graph Property, Independent Transversals, Vertex Partition