Edge‐partitions of planar graphs and their game coloring numbers

Let G be a planar graph and let g(G) and Δ(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) Δ(H) ≤ 4 if g(G) ≥ 5; (ii) Δ(H) ≤ 2 if g(G) ≥ 7; (iii) Δ(H)≤ 1 if g(G) ≥ 11; (iv) Δ(H) ≤ 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) ≤ 8 if g(G) ≥ 5; (ii) col_g(G)≤ 6 if g(G) ≥ 7; (iii) col_g(G) ≤ 5 if g(G) ≥ 11; (iv) col_g(G) ≤ 11 if G does not contain 4‐cycles (though it may contain 3‐cycles). © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 307–317, 2002