Planar Graph Coloring with an Uncooperative Partner

  title={Planar Graph Coloring with an Uncooperative Partner},
  author={Hal A. Kierstead and William T. Trotter},
  booktitle={Planar Graphs},
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… 

Figures from this paper

Game Chromatic Number of Graphs
y Abstract We show that if a graph has acyclic chromatic number k, then its game chromatic number is at most k(k + 1). By applying the known upper bounds for the acyclic chromatic numbers of various
The Two-Coloring Number and Degenerate Colorings of Planar Graphs
The two-coloring number of graphs, which was originally introduced in the study of the game chromatic number, also gives an upper bound on the degenerate chromatic number as introduced by Borodin. It
2-Coloring number revisited
Game coloring the Cartesian product of graphs
This article proves the following result: If G∗ has game coloring number m and G′ has acyclic chromatic number k, then the Cartesian product G G″ has game chromaticNumber at most k(k+m − 1).
Competitive Colorings of Oriented Graphs
This paper combines their technique with concepts introduced by several authors in a series of papers on game chromatic number to show that for every positive integer k,t here exists an integer t so that if C is a topologically closed class of graphs and C does not contain a complete graph on k vertices, then whenever G is an orientation of a graph from C, the oriented game Chromatic number of G is at most t.
On the Oriented Game Chromatic Number
It is proved that every oriented path has oriented game chromatic number at most 7 (and this bound is tight) and that there exists a constant t such thatevery oriented outerplanar graph has orientedGame chromaticNumber at most t.
Graph colorings with local constraints - a survey
  • Z. Tuza
  • Mathematics
    Discuss. Math. Graph Theory
  • 1997
This work surveys the literature on those variants of the chromatic number problem where not only a proper coloring has to be found but some further local restrictions are imposed on the color assignment.
The game of arboricity


On the Complexity of Some Coloring Games
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
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