Quadrangular embeddings of complete graphs and the Even Map Color Theorem (with details)

• Published in: arXiv.org
• Main Authors: Liu, Wenzhong, Lawrencenko, Serge, Chen, Beifang, Ellingham, M N, Hartsfield, Nora, Yang, Hui, Ye, Dong, Zha, Xiaoya
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 22.07.2021
• Subjects: Color; Embedding; Graphs; Mathematics - Combinatorics; Sharpness; Theorems
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1606.00948

Hartsfield and Ringel constructed orientable quadrangular embeddings of the complete graph \(K_n\) for \(n\equiv 5 \pmod 8\), and nonorientable ones for \(n \ge 9\) and \(n\equiv 1 \pmod 4\). These provide minimal quadrangulations of their underlying surfaces. We extend these results to determine, for every complete graph \(K_n\), \(n \ge 4\), the minimum genus, both orientable and nonorientable, for the surface in which \(K_n\) has an embedding with all faces of degree at least \(4\), and also for the surface in which \(K_n\) has an embedding with all faces of even degree. These last embeddings provide sharpness examples for a result of Hutchinson bounding the chromatic number of graphs embedded with all faces of even degree, completing the proof of the Even Map Color Theorem. We also show that if a connected simple graph \(G\) has a perfect matching and a cycle then the lexicographic product \(G[K_4]\) has orientable and nonorientable quadrangular embeddings; this provides new examples of minimal quadrangulations.