### Abstract

It was conjectured in 1981 by the third author that if a graph G does not contain more than t pairwise edge-disjoint triangles, then there exists a set of at most 2t edges that shares an edge with each triangle of G. In this paper, we prove this conjecture for odd-wheel-free graphs and for 'triangle-3-colorable' graphs, where the latter property means that the edges of the graph can be colored with three colors in such a way that each triangle receives three distinct colors on its edges. Among the consequences we obtain that the conjecture holds for every graph with chromatic number at most four. Also, two subclasses of K _{4}-free graphs are identified, in which the maximum number of pairwise edge-disjoint triangles is equal to the minimum number of edges covering all triangles. In addition, we prove that the recognition problem of triangle-3-colorable graphs is intractable.

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

Pages (from-to) | 381-392 |

Number of pages | 12 |

Journal | Graphs and Combinatorics |

Volume | 28 |

Issue number | 3 |

DOIs | |

Publication status | Published - May 2012 |

### Fingerprint

### Keywords

- Odd-wheel-free graphs
- T-transversal
- Triangle-3-colorable graphs

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Graphs and Combinatorics*,

*28*(3), 381-392. https://doi.org/10.1007/s00373-011-1048-8

**Small Edge Sets Meeting all Triangles of a Graph.** / Aparna Lakshmanan, S.; Bujtás, Cs; Tuza, Z.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 28, no. 3, pp. 381-392. https://doi.org/10.1007/s00373-011-1048-8

}

TY - JOUR

T1 - Small Edge Sets Meeting all Triangles of a Graph

AU - Aparna Lakshmanan, S.

AU - Bujtás, Cs

AU - Tuza, Z.

PY - 2012/5

Y1 - 2012/5

N2 - It was conjectured in 1981 by the third author that if a graph G does not contain more than t pairwise edge-disjoint triangles, then there exists a set of at most 2t edges that shares an edge with each triangle of G. In this paper, we prove this conjecture for odd-wheel-free graphs and for 'triangle-3-colorable' graphs, where the latter property means that the edges of the graph can be colored with three colors in such a way that each triangle receives three distinct colors on its edges. Among the consequences we obtain that the conjecture holds for every graph with chromatic number at most four. Also, two subclasses of K 4-free graphs are identified, in which the maximum number of pairwise edge-disjoint triangles is equal to the minimum number of edges covering all triangles. In addition, we prove that the recognition problem of triangle-3-colorable graphs is intractable.

AB - It was conjectured in 1981 by the third author that if a graph G does not contain more than t pairwise edge-disjoint triangles, then there exists a set of at most 2t edges that shares an edge with each triangle of G. In this paper, we prove this conjecture for odd-wheel-free graphs and for 'triangle-3-colorable' graphs, where the latter property means that the edges of the graph can be colored with three colors in such a way that each triangle receives three distinct colors on its edges. Among the consequences we obtain that the conjecture holds for every graph with chromatic number at most four. Also, two subclasses of K 4-free graphs are identified, in which the maximum number of pairwise edge-disjoint triangles is equal to the minimum number of edges covering all triangles. In addition, we prove that the recognition problem of triangle-3-colorable graphs is intractable.

KW - Odd-wheel-free graphs

KW - T-transversal

KW - Triangle-3-colorable graphs

UR - http://www.scopus.com/inward/record.url?scp=84860221541&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84860221541&partnerID=8YFLogxK

U2 - 10.1007/s00373-011-1048-8

DO - 10.1007/s00373-011-1048-8

M3 - Article

AN - SCOPUS:84860221541

VL - 28

SP - 381

EP - 392

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 3

ER -