Efficient graph packing via game Colouring

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 G₁ into the complement of G₂. We improve this by showing that if (gcol(G₁)1) Δ (G₂)+(gcol(G ₂)1)(G₁) < n then G₁ and G₂ pack. To our knowledge this is the first application of colouring games to a non-game problem.