### 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 language | English |
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Pages (from-to) | 333-351 |

Number of pages | 19 |

Journal | Combinatorica |

Volume | 26 |

Issue number | 3 |

DOIs | |

Publication status | Published - jún. 1 2006 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

### Cite this

*Combinatorica*,

*26*(3), 333-351. https://doi.org/10.1007/s00493-006-0019-9