Cubic graphs whose average number of regions is small

Some previously investigated infinite families of cubic graphs have the property that the average number of regions of a randomly selected orientable embedding is proportional to the number of their vertices. This paper demonstrates that this property is not true of connected graphs in general. That is, for every sufficiently large even value of n, there is an n-vertex cubic graph Gn with fewer than 1 + ln (n + 2) regions in its random orientable embedding. The proof provided is existential and no large cubic graphs are known that satisfy this scarceness of regions. It is conjectured that the complete graphs have a similar logarithmic bound and some numerical evidence is offered in support.