### Abstract

The game colouring number gcol(G) of a graph G is the least k such that, if two players take turns choosing the vertices of a graph, then either of them can ensure that every vertex has fewer than k neighbours chosen before it, regardless of what choices the other player makes. Clearly gcol(G) (G)+1. Sauer and Spencer [20] proved that if two graphs G1 and G2 on n vertices satisfy 2Δ (G1)≤ Δ(G2) < n then they pack, i.e., there is an embedding of G_{1} into the complement of G_{2}. We improve this by showing that if (gcol(G_{1})1) Δ (G_{2})+(gcol(G _{2})1)(G_{1}) < n then G_{1} and G_{2} pack. To our knowledge this is the first application of colouring games to a non-game problem.

Original language | English (US) |
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Pages (from-to) | 765-774 |

Number of pages | 10 |

Journal | Combinatorics Probability and Computing |

Volume | 18 |

Issue number | 5 |

DOIs | |

State | Published - Sep 1 2009 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics

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## Cite this

*Combinatorics Probability and Computing*,

*18*(5), 765-774. https://doi.org/10.1017/S0963548309009973