Decompositions of graphs of nonnegative characteristic with some forbidden subgraphs

A {\em $(d,h)$-decomposition} of a graph $G$ is an order pair $(D,H)$ such that $H$ is a subgraph of $G$ where $H$ has the maximum degree at most $h$ and $D$ is an acyclic orientation of $G-E(H)$ of maximum out-degree at most $d$. A graph $G$ is {\em $(d, h)$-decomposable} if $G$ has a $(d,h)$-decomposition. Let $G$ be a graph embeddable in a surface of nonnegative characteristic. In this paper, we prove the following results. (1) If $G$ has no chord $5$-cycles or no chord $6$-cycles or no chord $7$-cycles and no adjacent $4$-cycles, then $G$ is $(3,1)$-decomposable, which generalizes the results of Chen, Zhu and Wang [Comput. Math. Appl, 56 (2008) 2073--2078] and the results of Zhang [Comment. Math. Univ. Carolin, 54(3) (2013) 339--344]. (2) If $G$ has no $i$-cycles nor $j$-cycles for any subset $\{i,j\}\subseteq \{3,4,6\}$ is $(2,1)$-decomposable, which generalizes the results of Dong and Xu [Discrete Math. Alg. and Appl., 1(2) (2009), 291--297].