The chromatic number of a graph of girth 5 on a fixed surface.

*(English)*Zbl 1020.05030In this interesting paper it is proved that, for every surface \(S\) and every natural number \(k\), there exists a natural number \(f(S,k)\) such that the following holds: If \(G\) is a graph of girth 5 on \(S\), and \(H\) is a 3-colored subgraph with at most \(k\) vertices, then either the coloring of \(H\) can be extended to a 3-coloring of \(G\), or else there is a small obstruction containing \(H\), that is, a subgraph \(H'\) with at most \(f(S,k)\) vertices such that the coloring of \(H\) cannot be extended to a 3-coloring of \(H'\). In particular, there are only finitely many 4-color-critical graphs of girth 5 on \(S\), as a 4-color-critical graph of girth 5 on \(S\) has at most \(f(S,1)\) vertices. It follows that, if \(G\) is a graph of girth 5 on \(S\), and all noncontractible cycles in \(G\) have length greater than \(f(S,1)\), then \(G\) is 3-colorable. The result is best possible in the sense that there are infinitely many 4-color-critical graphs of girth 4 on \(S\), except when \(S\) is the sphere. As a consequence, it is deduced that the chromatic number of graphs of girth 5 on \(S\) can be found in polynomial time.

Reviewer: Ioan Tomescu (Bucureşti)

##### MSC:

05C15 | Coloring of graphs and hypergraphs |

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\textit{C. Thomassen}, J. Comb. Theory, Ser. B 87, No. 1, 38--71 (2003; Zbl 1020.05030)

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##### References:

[1] | Bondy, J.A.; Murty, U.S.R., Graph theory with applications, (1976), Macmillan London · Zbl 1134.05001 |

[2] | Fisk, S.; Mohar, B., Coloring graphs without short non-bounding cycles, J. combin. theory ser. B, 60, 268-276, (1994) · Zbl 0793.05058 |

[3] | Gimbel, J.; Thomassen, C., Coloring graphs with fixed genus and girth, Trans. amer. math. soc., 349, 4555-4564, (1997) · Zbl 0884.05039 |

[4] | Jensen, T.; Toft, B., Graph coloring problems, (1995), Wiley New York · Zbl 0855.05054 |

[5] | Malnic̆, A.; Mohar, B., Generating locally cyclic triangulation of surfaces, J. combin. theory ser. B, 56, 147-164, (1992) · Zbl 0723.05053 |

[6] | Mohar, B.; Thomassen, C., Graphs on surfaces, (2001), Johns Hopkins University Press Baltimore · Zbl 0979.05002 |

[7] | Thomassen, C., Every planar graph is 5-choosable, J. combin. theory ser. B, 62, 180-181, (1994) · Zbl 0805.05023 |

[8] | Thomassen, C., Grötzsch’s 3-color theorem and its counterpart for the torus and the projective plane, J. combin. theory ser. B, 62, 268-279, (1994) · Zbl 0817.05024 |

[9] | Thomassen, C., 3-List-coloring planar graphs of girth 5, J. combin. theory ser. B, 64, 101-107, (1995) · Zbl 0822.05029 |

[10] | Thomassen, C., Color-critical graphs on a fixed surface, J. combin. theory ser. B, 70, 67-100, (1997) · Zbl 0883.05051 |

[11] | C. Thomassen, A short list color proof of Grötzsch’s theorem, J. Combin. Theory Ser. B, to appear. |

[12] | B. Walls, Coloring girth restricted graphs on surfaces, Ph.D. Thesis, Georgia Institute of Technology, 1999. |

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