  Given a graph $G$, its genus polynomial is $\Gamma_G(x) = \sum_{k\geq 0}
g_k(G)x^k$, where $g_k(G)$ is the number of 2-cell embeddings of $G$ in an
orientable surface of genus $k$. The Log-Concavity Genus Distribution (LCGD)
Conjecture states that the genus polynomial of every graph is log-concave. It
was further conjectured by Stahl that the genus polynomial of every graph has
only real roots, however this was later disproved. We identify several examples
of cubic graphs whose genus polynomials, in addition to having at least one
non-real root, have a quadratic factor that is non-log-concave when factored
over the real numbers.
