How to Eliminate Crossings by Adding Handles or Crosscaps
نویسنده
چکیده
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 0 ; c 1 ; c 2 ; : : : encodes the trade-oo between adding handles and decreasing crossings. We focus on sequences of the type c 0 > c 1 > c 2 = 0; equivalently, we study the planar and toroidal crossing number of doubly-toroidal graphs. For every > 0 we construct graphs whose orientable crossing sequence satisiies c 1 =c 0 > 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 5 times more crossings. We similarly deene the non-orientable crossing sequence ~ c 0 ; ~ c 1 ; ~ c 2 ; : : : for drawings on non-orientable surfaces. We show that for every ~ c 0 > ~ c 1 > 0 there exists a graph with non-orientable crossing sequence ~ c 0 ; ~ c 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 diierent graphs).
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