  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4-genus of a complete graph, submitted to Discrete Mathematics and withdrawn by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.
