Determination of the 4-genus of a complete graph (with an appendix)

Abstract

In this paper, the quadrangular genus (4-genus) of the complete graph $K_p$ is shown to be $\gamma_4 (K_p) = \lceil {p(p-5)}/{8} \rceil +1$ for orientable surfaces. This means that $K_p$ is minimally embeddable in the closed orientable surface of genus $\gamma_4 (K_p)$ under the constraint that each face has length at least 4. In the most general setting, the genus of the complete graph was established by Ringel and Youngs and was mainly concerned with triangulations of surfaces. Nonetheless, since then a great deal of interest has also been generated in quadrangulations of surfaces. Hartsfield and Ringel were the first who considered minimal quadrangulations of surfaces. Sections 1--4 of this paper are essentially a reproduction of the original 1998 version as follows: Chen B., Lawrencenko S., Yang H. Determination of the 4-genus of a complete graph, submitted to Discrete Mathematics and withdrawn by S. Lawrencenko, June 1998, URL: https://t.co/cUg6R9Jwyw . More discussion on this 1998 version is held and some copyright issues around the quadrangular genus of complete graphs are clarified in the Appendix to the current version of the paper; the Appendix was written in 2017.