The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. Our main result is that every graph ▫$G$▫ that does not contain a fixed graph as a minor has ...crossing number ▫${\mathcal O}(\Delta n)$▫, where ▫$G$▫ has ▫$n$▫ vertices and maximum degree ▫$\Delta$▫. This dependence on ▫$n$▫ and ▫$\Delta$▫ is best possible. This result answers an open question of Wood and Telle [{New York J.~Mathematics}, 2007], who proved the best previous bound of ▫${\mathcal O}(\Delta^2 n)$▫. In addition, we prove that every ▫$K_5$▫-minor-free graph ▫$G$▫ has crossing number at most ▫$2\sum_v\deg(v)^2$▫, which again is the best possible dependence on the degrees of ▫$G$▫. We also study the convex and rectilinear crossing numbers, and prove an ▫${\mathcal O}(\Delta n)$▫ bound for the convex crossing number of bounded pathwidth graphs, and a ▫$\sum_v\deg(v)^2$▫ bound for the rectilinear crossing number of ▫$K_{3,3}$▫-minor-free graphs.
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