This paper investigates the maximum genus and upper embeddability of connected 4-regular graphs. We obtain lower bounds on the maximum genus of connected 4-regular simple graphs and connected 4-regular graphs without loops in terms of the Betti number. The definition of the Betti number is referred to [Gross and Tucker, Topological Graph Theory, New York, 1987]. Furthermore, we give examples that show that these lower bounds are tight.
ZHOU, DING; HAO, RONGXIA; and HE, WEILI
"Lower bounds for the maximum genus of 4-regular graphs,"
Turkish Journal of Mathematics: Vol. 36:
4, Article 3.
Available at: https://journals.tubitak.gov.tr/math/vol36/iss4/3