20.500.12556/RUL-177835 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 [Genus distributions for bouquets of circles, J. Combin. Theory Ser. B 47 (1989), 292-306] proposed a conjecture that the sequence $a_0(G), a_1(G), a_2(G), \dots$ 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., except that it is closer to the notion of $\Delta$-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. graphs on surfaces 2-cell embedding genus distribution log-concavity Log Concavity Conjecture true false true Angleški jezik Angleški jezik Članek v reviji 2026-01-09 09:16:53 2026-01-09 09:16:57 2026-01-10 04:27:53 0000-00-00 00:00:00 2026 0 0 Str. 164-179 Vol. 178 May 2026 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2026-05-01 519.17 0095-8956 10.1016/j.jctb.2025.12.005 264051715 1-s2.0-S0095895625001108-main.pdf 1-s2.0-S0095895625001108-main.pdf 1 D1BC008B2279B444B0980FA18CE725DD f3d1fe0a2ff410b34dc56a6621246f131a64c29305004d768752bc26e56fbe8a d7a8e602-ed33-11f0-bf92-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=225185 https://www.sciencedirect.com/science/article/pii/S0095895625001108 1 8e976950-ed33-11f0-bf92-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=225182 Fakulteta za matematiko in fiziko 0 0 0