There are fascinating combinatorial problems arising from that study. One of them is to reduce an important part of the problem to the study of the subset of the symmetric group that leads to complete leftward walks on a monograph. Another is to understand minimal complete leftward walks, that is to say those complete walks that visit only once each edge. Another one, quite difficult, is to see if we can produce all optimal examples of graphs in a surface of genus g with maximal valence and a complete walk, form sequences of blow-ups from the optimal monograph.
You can learn more about Professor Lalonde here.