Decompositions of graphs of nonnegative characteristic with some forbidden subgraphs

A (d,h)-decomposition of a graph G is an order pair (D,H) such that H subgraph where has the maximum degree at most h and D acyclic orientation G−E(H) out-degree d. (d,h)-decomposable if (d,h)-decomposition. Let be embeddable in surface nonnegative characteristic. It known (d,h)-decomposable, then h-defective d+1-choosable. In this paper, we investigate graphs prove following four results. (1) If no chord 5-cycles or 6-cycles, (3,1)-decomposable, which generalizes result Chen, Zhu Wang [Comput. Math. Appl. 56 (2008) 2073–2078]. (2) 7-cycles, (4,1)-decomposable, Zhang [Comment. Univ. Carolin, 54 (2013) 339–344]. (3) adjacent 3-cycles 6-cycles l-cycles for l∈{5,7}, (2,1)-decomposable, improves Xu Yu [Util. Math., 76 183–189]. (4) 4-cycles 8-cycles, Li, Lu, [Discrete Applied 331 (2023) 147–158]. We also show some these results are sharp.