A221222 Threshold for the P(n)-avoidance vertex-coloring game with two colors and fixed tree size restriction, where P(n) denotes the path on n vertices (see the comments and reference for precise definition).

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 791, 841, 902, 961, 1040, 1089, 1156, 1225, 1323, 1376, 1449, 1521, 1641, 1699, 1796, 1856, 1991, 2057, 2160, 2225, 2378, 2447, 2563, 2633, 2795, 2873

Offset: 1

Author

Torsten Muetze, Feb 22 2013

Keywords

• nonn
• hard

Comments

The game is played on a vertex-colored graph by two players called Builder and Painter. The game starts with an empty graph, i.e., no vertices are present in the beginning. In each step, Builder adds one new vertex to the graph, and a number of edges leading from previous vertices to this new vertex. Builder must not create any cycles, and all components (=trees) may have at most k vertices (k is fixed throughout the game). Painter immediately and irrevocably colors each new vertex red or blue. Painter's goal is to avoid creating a monochromatic (i.e., completely red or completely blue) path P(n) on n vertices (n is also fixed throughout the game). Builder's goal is to force Painter to create such a monochromatic P(n). a(n) is defined as the minimal k for which Builder wins this P(n)-avoidance game with two colors and tree size restriction k.

a(n)=n^2 exactly for all n in the set {1,...,27} U {29,31,33,34,35,39}, and a(n)>n^2 otherwise.

There are constants c_1 and c_2 such that c_1*n^2.01 <= a(n) <= c_2*n^2.59.

Examples

			For n=28, we have a(28)=791=28^2+7, meaning that the P(28)-avoidance game with two colors and tree size restriction k is a win for Builder for all k>=791, and a win for Painter for all k<791.
		

Links

• T. Mütze and R. Spöhel, On the path-avoidance vertex-coloring game, Electronic Journal of Combinatorics, 18(1) (2011), Research Paper 163, 33 pages.

Extensions

a(51)-a(53) from Torsten Muetze, Apr 22 2014