## Abstract

A graph G is k-critical if it has chromatic number k, but every proper subgraph of G is (k-1)-colorable. Let f_{k}(n) denote the minimum number of edges in an n-vertex k-critical graph. In a very recent paper, we gave a lower bound, f_{k}(n)≥(k, n), that is sharp for every n≡1 (mod k-1). It is also sharp for k=4 and every n≥6. In this note, we present a simple proof of the bound for k=4. It implies the case k=4 of two conjectures: Gallai in 1963 conjectured that if n≡1 (mod k-1) then (Formula presented), and Ore in 1967 conjectured that for every k≥4 and (Formula presented). We also show that our result implies a simple short proof of Grötzsch's Theorem that every triangle-free planar graph is 3-colorable.

Original language | English (US) |
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Pages (from-to) | 323-329 |

Number of pages | 7 |

Journal | Combinatorica |

Volume | 34 |

Issue number | 3 |

DOIs | |

State | Published - Jun 2014 |

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics