Efficient Graph Packing via Game Coloring

The game coloring 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 insure that every vertex has less than k neighbors 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 2∆(G1)∆(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)∆(G2) + (gcol(G2)− 1)∆(G1) < n then G1 and G2 pack. To our knowledge this is the first application of coloring games to a non-game problem.