 
Summary: TRADING CROSSINGS FOR HANDLES AND CROSSCAPS
DAN ARCHDEACON, C. PAUL BONNINGTON, AND JOZEF SIR
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Abstract. 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 ; c1 ; c2 ; : : : encodes the tradeo 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 doublytoroidal graphs. For
every > 0 we construct graphs whose orientable crossing sequence satises
c1 =c0 > 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 dene the nonorientable crossing sequence ~
c 0 ; ~ c1 ; ~ c2 ; : : : for
drawings on nonorientable surfaces. We show that for every ~ c 0 > ~
c 1 > 0 there
exists a graph with nonorientable crossing sequence ~
c 0 ; ~ c1 ; 0. We conjecture
