Date of Award
Honors Bachelor of Arts
German mathematician Claus Michael Ringel used voltage graphs to embed complete graphs onto orientable surfaces such that none of the graph's edges cross each other. Cayley maps do the same whilst being simpler to work with. The goal is to determine the efficiency of Cayley maps in embedding complete graphs onto orientable surfaces. This article focus on complete graphs of even order with an emphasis on graphs whose orders are congruent to 6 modulo 12 and 0 modulo 12. We establish 12 distinct classes that each have their own unique qualities. Through the generalization of a previous technique, we prove a nontrivial bound on the Cayley genus of graphs whose order is congruent to 6 modulo 12. We also show that Cayley maps cannot embed a complete graph onto its optimal genus for 8 out of the 12 classes provided the graph's order is greater than 6.
O'Connor, Michael, "Cayley Map Embeddings of Complete Graphs With Even Order" (2023). Honors Program Theses. 204.
Michael Wayne O'Connor