Trading crossings for handles and crosscaps

Let c_k = cr_k (G) denote the minimum number of edge crossings when a graph G is drawn on an orientable surface of genus k. The (orientable) crossing sequence c_o, c₁,c₂…encodes the trade‐off between adding handles and decreasing crossings. We focus on sequences of the type c_o > c₁ > c₂ = 0; equivalently, we study the planar and toroidal crossing number of doubly‐toroidal graphs. For every ? > 0 we construct graphs whose orientable crossing sequence satisfies c₁/c_o > 5/6??. In other words, we construct graphs where the addition of one handle can save roughly 1/6th of the crossings, but the addition of a second handle can save five times more crossings. We similarly define the non‐orientable crossing sequence ?₀,?₁,?₂, ··· for drawings on non‐orientable surfaces. We show that for every ?₀ > ?₁ > 0 there exists a graph with non‐orientable crossing sequence ?₀, ?₁, 0. We conjecture that every strictly‐decreasing sequence of non‐negative integers can be both an orientable crossing sequence and a non‐orientable crossing sequence (with different graphs). © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 230–243, 2001