Given a graph G, its genus distribution is the sequence whose k-th term is the number of 2-cell embeddings of G in the orientable surface of genus k. The Log-Concavity Genus Distribution (LCGD) Conjecture states that the genus distribution of every graph is log-concave. This has been proven to hold for several classes of graphs, including ladders, bouquets, and dipoles, but was recently shown by Mohar to be false for 4-connected graphs. Using local rotations and facial walks, we provide some evidence to support the LCGD Conjecture for cubic cyclically 5-edge-connected planar graphs by showing that the log-concavity condition holds for the initial terms of the genus distribution. Additionally, we demonstrate how the dual graph can provide insight into the facial structure of an embedding of a cubic graph.
This is joint work with Bojan Mohar (Simon Fraser University).