Log-concavity of combinations of sequences and applications to genus distributions

Jonathan L. Gross, Toufik Mansour, Thomas W. Tucker, David G.L. Wang

Research output: Contribution to journalArticlepeer-review

Abstract

We formulate conditions on a set of log-concave sequences, under which any linear combination of those sequences is log-concave, and further, of conditions under which linear combinations of log-concave sequences that have been transformed by convolution are log-concave. These conditions involve relations on sequences called synchronicity and ratio-dominance, and a characterization of some bivariate sequences as lexicographic. We are motivated by the 25-year-old conjecture that the genus distribution of every graph is log-concave. Although calculating genus distributions is NP-hard, they have been calculated explicitly for many graphs of tractable size, and the three conditions have been observed to occur in the partitioned genus distributions of all such graphs. They are used here to prove the log-concavity of the genus distributions of graphs constructed by iterative amalgamation of double-rooted graph fragments whose genus distributions adhere to these conditions, even though it is known that the genus polynomials of some such graphs have imaginary roots. A blend of topological and combinatorial arguments demonstrates that log-concavity is preserved through the iterations.

Original languageEnglish
Pages (from-to)1002-1029
Number of pages28
JournalSIAM Journal on Discrete Mathematics
Volume29
Issue number2
DOIs
StatePublished - 2015

Bibliographical note

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Keywords

  • Genus distribution
  • Log-concavity

ASJC Scopus subject areas

  • Mathematics (all)

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