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Full-Text Articles in Discrete Mathematics and Combinatorics
Cayley Map Embeddings Of Complete Graphs With Even Order, Michael O'Connor
Cayley Map Embeddings Of Complete Graphs With Even Order, Michael O'Connor
Honors Program Theses
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 …
Finding Triangular Cayley Maps With Graph Touring, Hannah Hendrickson
Finding Triangular Cayley Maps With Graph Touring, Hannah Hendrickson
Honors Program Theses
We develop a method for determining whether certain kinds of Cayley maps can exist by using multi-digraph representations of the data in the Cayley maps. Euler tours of these multi-digraphs correspond exactly to the permutations which define Cayley maps. We also begin to classify which 3-regular multi-digraphs have "non-backtracking" Euler tours in general.
Cayley Map Embeddings Of Complete Graphs, Miriam Scheinblum
Cayley Map Embeddings Of Complete Graphs, Miriam Scheinblum
Honors Program Theses
This paper looks at Cayley map embeddings of complete graphs on orientable surfaces. Cayley maps constrain graph embeddings to those with cyclical edge rotations, so optimal embeddings on surfaces with the minimum genus may not always be possible. We explore instances when Cayley maps succeed at optimally embedding complete graphs, and when optimal embeddings are not possible, we determine how close to optimal they can get by finding vertex rotations that result in the smallest possible genus. Many of the complete graphs we consider have prime numbers of vertices, so for each complete graph Kn we focus on mappings with …