Geodesic
Simultaneous current graph constructions for minimum triangulations and complete graph embeddings
Ars mathematica contemporanea, Tome 18 (2020) no. 2, pp. 309-337 Cet article a éte moissonné depuis la source Ars Mathematica Contemporanea website

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The problems of calculating the genus of the complete graphs and of finding a minimum triangulation 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 addition to Cases 6 and 9, in a unified manner, greatly simplifying 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.
DOI : 10.26493/1855-3974.2006.fa4
Keywords: Topological graph theory, current graphs, map coloring, triangulations 
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     author = {Timothy Sun},
     title = {
		{Simultaneous} current graph constructions for minimum triangulations and complete graph embeddings
	},
     journal = {Ars mathematica contemporanea},
     pages = {309--337},
     year = {2020},
     volume = {18},
     number = {2},
     doi = {10.26493/1855-3974.2006.fa4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.2006.fa4/}
}
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Timothy Sun. Simultaneous current graph constructions for minimum triangulations and complete graph embeddings. Ars mathematica contemporanea, Tome 18 (2020) no. 2, pp. 309-337. doi: 10.26493/1855-3974.2006.fa4

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