Z2-genus of graphs and Hanani-Tutte theorems

Abstract

In this thesis, we present a counterexample to an extension of the Uni- fied Hanani-Tutte theorem to the Klein bottle, as well as to other variants that allow crossings only between dependent edges. We also use an existing method for bounding the Euler Z2-genus and orientable Z2-genus of Kn,m to prove a lower bound on these genera for Kn. These bounds differ from the corresponding classical Euler and orientable genera only in the linear term.

V tejto pra ́ci uva ́dzame protiprı ́klad k rozs ̌ı ́reniu zjednotenej Hananiho- Tutteovej vety na Kleinovu fl'as ̌u, ako aj k iny ́m variantom, ktore ́ povol'uju ́ krı ́z ̌enia iba medzi susedny ́mi hranami. D ̌alej vyuz ̌ı ́vame existuju ́cu meto ́du na odhad Eulerovho Z2-rodu a orientovatel'ne ́ho Z2-rodu pre Kn,m na do ̂kaz dolny ́ch odhadov ty ́chto rodov pre Kn. Tieto odhady sa od prı 'slus ̌ny ́ch kla- sicky ́ch Eulerovy ́ch a orientovatel'ny ́ch rodov lı 's ̌ia iba v linea ́rnom c ̌lene. 1