## Abstract

Graphs distinguished by K_{r}-minor prohibition limited to subgraphs induced by circuits have chromatic number bounded by a function f(r); precise bounds on f(r) are unknown. If minor prohibition is limited to subgraphs induced by simple paths instead of circuits, then for certain forbidden configurations, we reach tight estimates. A graph whose simple paths induce K_{3,3}-minor free graphs is proven to be 6-colorable; k_{5} is such a graph. Consequently, a graph whose simple paths induce planar graphs is 6-colorable. We suspect the latter to be 5-colorable and we are not aware of such 5-chromatic graphs. Alternatively, (and with more accuracy) a graph whose simple paths induce k_{5},K_{3,3}^{-} minor free graphs is proven to be 4-colorable (where K_{3,3}^{-} is the graph obtained from K_{3,3} by removing a single edge); K_{4} is such a graph.

Original language | English |
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Pages (from-to) | 699-704 |

Number of pages | 6 |

Journal | Discrete Mathematics |

Volume | 311 |

Issue number | 8-9 |

DOIs | |

State | Published - 6 May 2011 |

Externally published | Yes |

## Keywords

- Bridges of circuits
- Chromatic number