Removing Even Crossings on Surfaces

In this paper we investigate how certain results related to the HananiTutte theorem can be extended from the plane to surfaces. We give a simple topological proof that the weak Hanani-Tutte theorem is true on arbitrary surfaces, both orientable and nonorientable. We apply these results and the proof techniques to obtain new and old results about generalized thrackles, including that every bipartite generalized thrackle on a surface S can be embedded on S. We also extend to arbitrary surfaces a result of Pach and T´oth that allows the redrawing of a graph so as to remove all crossings with even edges. From this we can conclude that crS(G), the crossing number of a graph G on surface S, is bounded by 2 ocrS(G)2 , where ocrS(G) is the odd crossing number of G on surface S. Finally, we prove that ocrS(G) = crS(G) whenever ocrS(G) ≤ 2, for any surface S.

• Type: Journal
• Year: 2007
• Where: ENDM
• Authors: Michael J. Pelsmajer, Marcus Schaefer, Daniel Stefankovic
• Keywords: ENDM 2007, Simple Topological Proof, Surfaces, Weak Hanani-tutte Theorem