Strong log-convexity of genus sequences
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918264224874496 |
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| author | Mohar, Bojan |
| author_facet | Mohar, Bojan |
| contents | 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 $Δ$-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_10854 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strong log-convexity of genus sequences Mohar, Bojan Combinatorics 05C10 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 $Δ$-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. |
| title | Strong log-convexity of genus sequences |
| topic | Combinatorics 05C10 |
| url | https://arxiv.org/abs/2405.10854 |