Edge-partitions of graphs of nonnegative characteristic and their game coloring numbers

Let GG be a graph of nonnegative characteristic and let g(G)g(G) and Δ(G)Δ(G) be its girth and maximum degree, respectively. We show that GG has an edge-partition into a forest and a subgraph HH so that (1) Δ(H)⩽1Δ(H)⩽1 if g(G)⩾11g(G)⩾11; (2) Δ(H)⩽2Δ(H)⩽2 if g(G)⩾7g(G)⩾7; (3) Δ(H)⩽4Δ(H)⩽4 if either g(G)⩾5g(G)⩾5 or GG does not contain 4-cycles and 5-cycles; (4) Δ(H)⩽6Δ(H)⩽6 if GG does not contain 44-cycles. These results are applied to find the following upper bounds for the game coloring number colg(G)colg(G) of GG: (1) colg(G)⩽5colg(G)⩽5 if g(G)⩾11g(G)⩾11; (2) colg(G)⩽6colg(G)⩽6 if g(G)⩾7g(G)⩾7; (3) colg(G)⩽8colg(G)⩽8 if either g(G)⩾5g(G)⩾5 or GG contains no 4-cycles and 5-cycles; (4) colg(G)⩽10colg(G)⩽10 if GG does not contain 4-cycles.