  Recently, Dasbach, Futer, Kalfagianni, Lin, and Stoltzfus extended the notion
of a Tait graph by associating a set of ribbon graphs (or equivalently,
embedded graphs) to a link diagram. Here we focus on Seifert graphs, which are
the ribbon graphs of a knot or link diagram that arise from Seifert states. We
provide a characterization of Seifert graphs in terms of Eulerian subgraphs.
This characterization can be viewed as a refinement of the fact that Seifert
graphs are bipartite. We go on to examine the family of ribbon graphs that
arises by forming the parallels of a link diagram and determine how the genus
of the ribbon graph of a $r$-fold parallel of a link diagram is related to that
of the original link diagram.
