It is well known that finding the crossing number of a graph on nonplanar surfaces is very difficult.In this paper we study the crossing number of the circular graph C（10,4） on the projective plane and determine the nonorientable crossing number sequence of C（10,4）.On the basis of the result,we show that the nonorientable crossing number sequence of C（10,4） is not convex.
In this paper, we discuss the crossing numbers of two one-vertex maps on orientable surfaces. By using a reductive method, we give the crossing number of two one-vertex maps with one face on an orientable surface and the crossing number of a one-vertex map with one face and a one-vertex map with two faces on an orientable surface. This provides a lower bound for the crossing number of two general maps on an orientable surface.
In this paper we show that the face-width of any embedding of a Halin graph（a type of planar graph） in the torus is one, and give a formula for determining the number of all nonequivalent embeddings of a Halin graph in the torus.
In this paper, we show that for a locally LEW-embedded 3-connected graph G in orientable surface, the following results hold： 1） Each of such embeddings is minimum genus embedding; 2） The facial cycles are precisely the induced nonseparating cycles which implies the uniqueness of such embeddings; 3） Every overlap graph O（G, C） is a bipartite graph and G has only one C-bridge H such that C U H is nonplanar provided C is a contractible cycle shorter than every noncontractible cycle containing an edge of C. This extends the results of C Thomassen＇s work on LEW-embedded graphs.
In this paper, we investigate fundamental cycles in a graph G and their relations with graph embeddings. We show that a graph G may be embedded in an orientable surface with genus at least g if and only if for any spanning tree T, there exists a sequence of fundamental cycles C 1,C 2,…,C 2g with C 2i?1 ∩ C 2i ≠ /0 for 1 ? i ? g. In particular, among β(G) fundamental cycles of any spanning tree T of a graph G, there are exactly 2γM (G) cycles C 1, C 2,…,C 2γM(G) such that C 2i?1 ∩ C 2i ≠ /0 for 1 ? i ? γM (G), where β(G) and γM (G) are the Betti number and the maximum genus of G, respectively. This implies that it is possible to construct an orientable embedding with large genus of a graph G from an arbitrary spanning tree T (which may have very large number of odd components in G E(T)). This is different from the earlier work of Xuong and Liu, where spanning trees with small odd components are needed. In fact, this makes a common generalization of Xuong, Liu and Fu et al. Furthermore, we show that (1) this result is useful for locating the maximum genus of a graph having a specific edge-cut. Some known results for embedded graphs are also concluded; (2) the maximum genus problem may be reduced to the maximum matching problem. Based on this result and the algorithm of Micali-Vazirani, we present a new efficient algorithm to determine the maximum genus of a graph in $ O((\beta (G))^{\frac{5} {2}} ) $ steps. Our method is straight and quite different from the algorithm of Furst, Gross and McGeoch which depends on a result of Giles where matroid parity method is needed.
