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.

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Correspondence to Tibor Szabó*.

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* Research supported by the joint Berlin/Zurichgrad uate program Combinatorics, Geometry, Computation, financed by the German Science Foundation (DFG) and ETH Zürich.

† Research partially supported by Hungarian National Research Fund grants T-037846, T-046234 and AT-048826.

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Szabó*, T., Tardos†, G. Extremal Problems For Transversals In Graphs With Bounded Degree. Combinatorica 26, 333–351 (2006). https://doi.org/10.1007/s00493-006-0019-9

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Mathematics Subject Classification (2000):

  • 05D15
  • 05C15
  • 05C69