Combinatorial conjectures that imply local log-concavity of graph genus polynomials

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

Research output: Contribution to journalArticlepeer-review


The 25-year old LCGD Conjecture is that the genus distribution of every graph is log-concave. We present herein a new topological conjecture, called the Local Log-Concavity Conjecture. We also present a purely combinatorial conjecture, which we prove to be equivalent to the Local Log-Concavity Conjecture. We use the equivalence to prove the Local Log-Concavity Conjecture for graphs of maximum degree four. We then show how a formula of David Jackson could be used to prove log-concavity for the genus distributions of various partial rotation systems, with straight-forward application to proving the local log-concavity of additional classes of graphs. We close with an additional conjecture, whose proof, along with proof of the Local Log-Concavity Conjecture, would affirm the LCGD Conjecture.

Original languageEnglish
Pages (from-to)207-222
Number of pages16
JournalEuropean Journal of Combinatorics
StatePublished - 1 Feb 2016

Bibliographical note

Publisher Copyright:
© 2015 .

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics


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