The crossing number of a graph G, cr(G) is the minimum number of intersections among edges over all possible drawings on a plane. The pairwise... Show moreThe crossing number of a graph G, cr(G) is the minimum number of intersections among edges over all possible drawings on a plane. The pairwise crossing number pcr(G) is the the minimum number of pairs of edges that cross at least once over drawings. In the rst part of this survey, we deal with the conjecture that pcr(G) = cr(G), and prove that this is true for 4-edge weighted maps on the annulus. Moreover, we develop methods for solving analogous n-edge problems including the classi cation of permutations on a circle. In the second part, we de ne the generalized crossing number cri(G) as the crossing number of a graph on the orientable surface of genus i. The crossing sequence is de ned as (cri(G))g(G) i=0 , where g(G) is the genus of the graph. This part aims at the conjecture that for each sequence of four numbers decreasing to 0, there is some graph with such numbers as its crossing sequence. We come up with a particular family of graphs which have concave crossing sequences of length 4, but partially prove it. M.S. in Applied Mathematics, May 2013 Show less