Crossing Numbers on the Disk with Multiple Holes

In this paper we look at the crossing number, pair crossing number, and odd crossing number of graphs on a disk with multiple holes. We provide an upper bound ⌊ (n−1)2 4 ⌋ for the odd crossing number of graphs on the annulus with n edges. For graphs on a disk with multiple holes, we develop polynomial time algorithms to find the number of crossings between two edges, given their homotopy classes. We describe each edge as a word referring to its homotopy class. The algorithms also proved the crossing number, pair crossing number, and odd crossing number for graphs on a disk with multiple holes.