6 papers · 1 filter
Embeddings of critical graphs near the Heawood bound
Timothy Sun
Complementing a theorem of Å krekovski, we characterize the -critical graphs embeddable in surfaces of Euler genus at least , where denotes the Heawood number of the…
Revisiting Cases 2 and 11 of the Map Color Theorem
Timothy Sun
In 1968, Ringel and Youngs solved the remaining cases of the orientable Map Color Theorem by finding genus embeddings of the complete graphs , for sufficiently large $n \equiv…
Orientable triangular embeddings of the complete graphs on 36s vertices from noncyclic current graphs
Avinh Huynh, Timothy Sun
Mark Jungerman's 1975 Ph.D. thesis presents several infinite families of index 2 current graphs that generate triangular embeddings of complete or near-complete graphs. However, th…
Genus embeddings of complete graphs minus a matching
Timothy Sun
We show that for all , , there is an orientable triangular embedding of the octahedral graph on vertices that can be augmented with handles to p…
An optimal construction for complete graph embeddings with duals of low connectivity
Timothy Sun
We describe a construction for embeddings of complete graphs where the dual has a cutvertex and the genus is close to the minimum genus of the primal graph. When the number of vert…
On Kainen's conjectures on surface crossing numbers
Timothy Sun
In 1972, Kainen proved a general lower bound on the crossing number of a graph in a closed surface and conjectured that this bound is tight when the graph is either a complete grap…