Degree sequences for k-regulable ribbon realizations

Abstract

The motivation question as to which ribbon graphs have a 4-regular checkerboard colorable twual is posed by Ellis-Monaghan and Moffatt. The ribbon graph $G$ is a realization of the sequence $D$ if its degree sequence is $D$. Furthermore, we refer to $G$ as a $k$-regulable realization of $D$ if the realization $G$ of $D$ has a $k$-regular partial dual. Since any Eulerian ribbon graph has a checkerboard colorable partial Petrial, we attempt to distinguish the ribbon graphs with $k$-regular partial duals directly from their degree sequences. Although the result is frustrating, some sequences lacking $k$-regulable realization are exposed. Moreover, we construct a family of $k$-regulable realizations for all sequences whose elements are greater than 1, except for the sequence $(2,\dots ,2,3,\dots ,3,5)$ when $k=3$, where the number of 2s and 3s are $3j_1-1$ and $2j_2-1$, respectively, $j_1,j_2>0$.