The problems of the genus of the complete graphs and minimum triangulations for each surface were both solved using the theory of current graphs, and each of them divided into twelve different cases, depending on the residue modulo 12 of the number of vertices. Cases 8 and 11 were of particular difficulty for both problems, with multiple families of current graphs developed to solve these cases. We solve these cases in a unified manner with families of current graphs applicable to both problems. Additionally, we give new constructions to both problems for Cases 6 and 9, which greatly simplify previous constructions by Ringel, Youngs, Guy, and Jungerman. All these new constructions are index 3 current graphs sharing nearly all of the structure of the simple solution for Case 5 of the Map Color Theorem.

Rights

URN

URN:NBN:SI:DOC-MNEJA1KL

COBISSID

41170435

Added

28.04.2021

Metadata

Citation

APA:

Sun, Timothy (2020). Simultaneous current graph constructions for minimum triangulations and complete graph embeddings. Ars mathematica contemporanea, volume 18, issue 2, str. 309-337. URN:NBN:SI:DOC-MNEJA1KL from http://www.dlib.si

MLA:

Sun, Timothy. "Simultaneous current graph constructions for minimum triangulations and complete graph embeddings." Ars mathematica contemporanea volume 18. issue 2 (2020) str. 309-337.
<http://www.dlib.si/?URN=URN:NBN:SI:DOC-MNEJA1KL>

Ars mathematica contemporanea
2008-
(Assembly record)