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  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.
ASJC Scopus subject areas
- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics