Extremal problems for transversals in graphs with bounded degree

Tibor Szabó, G. Tardos

Research output: Contribution to journalArticle

22 Citations (Scopus)

Abstract

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+/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, Erdos 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.

Original languageEnglish
Pages (from-to)333-351
Number of pages19
JournalCombinatorica
Volume26
Issue number3
DOIs
Publication statusPublished - Jun 2006

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Transversals
Extremal Problems
Triangulation
Graph in graph theory
Vertex of a graph
Sperner's Lemma
Vertex Partition
Topological Methods
Erdös
Regular Graph
Maximum Degree
Connected Components
Subgraph
Disjoint

ASJC Scopus subject areas

  • Mathematics(all)
  • Discrete Mathematics and Combinatorics

Cite this

Extremal problems for transversals in graphs with bounded degree. / Szabó, Tibor; Tardos, G.

In: Combinatorica, Vol. 26, No. 3, 06.2006, p. 333-351.

Research output: Contribution to journalArticle

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