The Genus of a Graph: A Survey

The problem of determining the genus for a graph can be dated to Map Color Conjecture proposed by Heawood in 1890. This was implied Thread Problem Hilbert and Cohn-Vossen. conjecture finally established Ringel, Youngs, many other mathematicians. Subsequently, genera some special graphs with symmetry were determined. study embeddings is closely related invariants graph. Specifically, computational complexity dependent on underlying certain well-known NP-hard problems. In this survey, main construction techniques criteria are stated topic Most known listed. A new theorem shown that method joint trees reasonable. Moreover, formal set introduced, results obtained. Although cubic Hamilton cycle asymmetric, it interesting associate surfaces all its symmetry.