Archdeacon, D.; Bonnington, P. and Siran, J.
Trading crossings for handles and crosscaps.
Journal of Graph Theory, 38(4) pp. 230–243.
Let ck = crk (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 co, c1,c2encodes the trade-off between adding handles and decreasing crossings. We focus on sequences of the type co > 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/co > 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).
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