Let H, T and Cn​ be a graph, a tree and a cycle of order n,
respectively. Let H(i) be the complete join of H and an empty graph on
i vertices. Then the Cartesian product Hâ–¡T of H and T can be
obtained by applying zip product on H(i) and the graph produced by zip
product repeatedly. Let crΣ​(H) denote the crossing number of
H in an arbitrary surface Σ. If H satisfies certain connectivity
condition, then crΣ​(H□T) is not less than the sum of the
crossing numbers of its ``subgraphs". In this paper, we introduced a new
concept of generalized periodic graphs, which contains H□Cn​. For a
generalized periodic graph G and a function f(t), where t is the number
of subgraphs in a decomposition of G, we gave a necessary and sufficient
condition for crΣ​(G)≥f(t). As an application, we
confirmed a conjecture of Lin et al. on the crossing number of the generalized
Petersen graph P(4h+2,2h) in the plane. Based on the condition, algorithms
are constructed to compute lower bounds on the crossing number of generalized
periodic graphs in Σ. In special cases, it is possible to determine
lower bounds on an infinite family of generalized periodic graphs, by
determining a lower bound on the crossing number of a finite generalized
periodic graph.Comment: 26 pages, 20 figure