(1,{\lambda})-embedded graphs and the acyclic edge choosability
  A (1,{\lambda})-embedded graph is a graph that can be embedded on a surface with Euler characteristic {\lambda} so that each edge is crossed by at most one other edge. A graph G is called {\alpha}-linear if there exists an integral constant {\beta} such that e(G') \leq {\alpha} v(G')+{\beta} for each G'\subseteq G. In this paper, it is shown that every (1,{\lambda})-embedded graph G is 4-linear for all possible {\lambda}, and is acyclicly edge-(3{\Delta}(G)+70)-choosable for {\lambda}=1,2. 
Xin Zhang, Guizhen Liu, Jian-Liang Wu
math.CO cs.DM