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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 anorientable surface of genus $k$. The Log-Concavity Genus Distribution (LCGD)Conjecture states that the genus polynomial of every graph is log-concave. Itwas further conjectured by Stahl that the genus polynomial of every graph hasonly real roots, however this was later disproved. We identify several examplesof cubic graphs whose genus polynomials, in addition to having at least onenon-real root, have a quadratic factor that is non-log-concave when factoredover the real numbers.

Genus Polynomials of Cubic Graphs with Non-Real Roots