### Abstract

Motivated by a problem of Gallai on (1-1)-transversals of 2-intervals, it was proved by the authors in 1969 that if the edges of a complete graph K are colored with red and blue (both colors can appear on an edge) so that there is no monochromatic induced C_{4} and C_{5} then the vertices of K can be partitioned into a red and a blue clique. Aharoni, Berger, Chudnovsky and Ziani recently strengthened this by showing that it is enough to assume that there is no induced monochromatic C_{4} and there is no induced C_{5} in one of the colors. Here we strengthen this result further, showing that it is enough to assume that there is no monochromatic induced C_{4} and there is no K5 on which both color classes induce a C_{5}. We also answer a question of Kaiser and Rabinovich, giving an example of six 2-convex sets in the plane such that any three intersect but there is no (1-1)-transversal for them.

Original language | English |
---|---|

Journal | Electronic Journal of Combinatorics |

Volume | 23 |

Issue number | 3 |

Publication status | Published - Sep 2 2016 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Computational Theory and Mathematics

### Cite this

*Electronic Journal of Combinatorics*,

*23*(3).

**Red-blue clique partitions and (1-1)-transversals.** / Gyárfás, A.; Lehel, Jenő.

Research output: Contribution to journal › Article

*Electronic Journal of Combinatorics*, vol. 23, no. 3.

}

TY - JOUR

T1 - Red-blue clique partitions and (1-1)-transversals

AU - Gyárfás, A.

AU - Lehel, Jenő

PY - 2016/9/2

Y1 - 2016/9/2

N2 - Motivated by a problem of Gallai on (1-1)-transversals of 2-intervals, it was proved by the authors in 1969 that if the edges of a complete graph K are colored with red and blue (both colors can appear on an edge) so that there is no monochromatic induced C4 and C5 then the vertices of K can be partitioned into a red and a blue clique. Aharoni, Berger, Chudnovsky and Ziani recently strengthened this by showing that it is enough to assume that there is no induced monochromatic C4 and there is no induced C5 in one of the colors. Here we strengthen this result further, showing that it is enough to assume that there is no monochromatic induced C4 and there is no K5 on which both color classes induce a C5. We also answer a question of Kaiser and Rabinovich, giving an example of six 2-convex sets in the plane such that any three intersect but there is no (1-1)-transversal for them.

AB - Motivated by a problem of Gallai on (1-1)-transversals of 2-intervals, it was proved by the authors in 1969 that if the edges of a complete graph K are colored with red and blue (both colors can appear on an edge) so that there is no monochromatic induced C4 and C5 then the vertices of K can be partitioned into a red and a blue clique. Aharoni, Berger, Chudnovsky and Ziani recently strengthened this by showing that it is enough to assume that there is no induced monochromatic C4 and there is no induced C5 in one of the colors. Here we strengthen this result further, showing that it is enough to assume that there is no monochromatic induced C4 and there is no K5 on which both color classes induce a C5. We also answer a question of Kaiser and Rabinovich, giving an example of six 2-convex sets in the plane such that any three intersect but there is no (1-1)-transversal for them.

UR - http://www.scopus.com/inward/record.url?scp=84989853670&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84989853670&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84989853670

VL - 23

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 3

ER -