## Abstract

Let V be a vector space of dimension ν over a field of order q. The q-Kneser graph has the k-dimensional subspaces of V as its vertices, where two subspaces α and β are adjacent if and only if α ∩ β is the zero subspace. This paper is motivated by the problem of determining the chromatic numbers of these graphs. This problem is trivial when k = 1 (and the graphs are complete) or when ν < 2k (and the graphs are empty). We establish some basic theory in the general case. Then specializing to the case k = 2 , we show that the chromatic number is q^{2} + q when ν = 4 and (q^{ν-1}- 1)/(q-1) when ν > 4. In both cases we characterise the minimal colourings.

Original language | English (US) |
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Pages (from-to) | 39-52 |

Number of pages | 14 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 113 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2006 |

Externally published | Yes |

## Keywords

- Chromatic number
- Kneser graph
- Projective space