Strong log-convexity of genus sequences

For a graph G , and a nonnegative integer g , let a g ( G ) be the number of 2-cell embeddings of G in an orientable surface of genus g (counted up to the combinatorial homeomorphism equivalence). In 1989, Gross, Robbins, and Tucker [21] proposed a conjecture that the sequence a 0 ( G ) , a 1 ( G ) , a 2 ( G ) , … is log-concave for every graph G . This conjecture is reminiscent to the Heron–Rota–Welsh Log Concavity Conjecture that was recently resolved in the affirmative by June Huh et al. [24] , [1] , except that it is closer to the notion of Δ-matroids than to the usual matroids. In this short paper, we disprove the Log Concavity Conjecture of Gross, Robbins, and Tucker by providing examples that show strong deviation from log-concavity at multiple terms of their genus sequences.