Planar Graph Coloring with an Uncooperative Partner

@inproceedings{Kierstead1991PlanarGC,
  title={Planar Graph Coloring with an Uncooperative Partner},
  author={Hal A. Kierstead and William T. Trotter},
  booktitle={Planar Graphs},
  year={1991}
}
We show that the game chromatic number of a planar graph is at most 33. More generally, there exists a function f: f\l --+ f\l so that for each n E f\l. if a graph does not contain a homeomorph of Kn• then its game chromatic number is at most f(n). In particular, the game chromatic number of a graph is bounded in terms of its genus. Our proof is motivated by the concept of p-arrangeability, which was first introduced by Guantao and Schelp in a Ramsey theoretic setting. © 1994 John Wiley & Sons… 

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References

SHOWING 1-9 OF 9 REFERENCES
On the Complexity of Some Coloring Games
TLDR
It is shown that for both variants of the game, the problem of determining whether there is a winning strategy for player 1 is PSPACE-complete for any C with |C|≥3, but the problems are solvable in , and time, respectively, if |C |=2.
Graphs with Linearly Bounded Ramsey Numbers
TLDR
It is proved that for each p ≥ 1, there is a constant c (depending only on p) such that the Ramsey number r ( G, G ) ≤ cn for eachp -arrangeable graph G of order n.
Regular Partitions of Graphs
Abstract : A crucial lemma in recent work of the author (showing that k-term arithmetic progression-free sets of integers must have density zero) stated (approximately) that any large bipartite graph
On sets of integers containing k elements in arithmetic progression
In 1926 van der Waerden [13] proved the following startling theorem : If the set of integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmetic
On the game chromatic number of some classes of graphs
On the magnitude of generalized ramsey numbers. Infinite and Finite Sets
  • Colloquium of the Mathematics Society Janos Bolyai
The Ramsey number of a graph of bounded degree
  • J. Combinat. Theory B
  • 1983