Edge-partitions of planar graphs and their game coloring numbers

Let G be a planar graph and let g(G) and Delta(G) be its girth and maximum degree, respectively. We show that G has an edge-partition into a forest and a subgraph H so that (i) Delta(H) less than or equal to 4 if g(G) greater than or equal to 5; (ii) Delta(H) less than or equal to 2 if g(G) greater than or equal to 7; (iii) Delta(H) less than or equal to 1 if g(G) greater than or equal to 11; (iv) Delta(H) less than or equal to 7 if G does not contain 4-cycles (though it may contain 3-cycles). These results are applied to find the following upper bounds for the game coloring number col(g)(G) of a planar graph G: (i) col(g)(G) less than or equal to 8 if g(G) > 5; (ii) col(g)(G) less than or equal to 6 if g(G) greater than or equal to 7; (iii) col(g)(G) less than or equal to 5 if g(G) greater than or equal to 11; (iv) col(g)(G) less than or equal to 11 if G does not contain 4-cycles (though it may contain 3-cycles). (C) 2002 Wiley Periodicals, Inc.