Abstract
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 language | English |
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Pages (from-to) | 207-222 |
Number of pages | 16 |
Journal | European Journal of Combinatorics |
Volume | 52 |
DOIs | |
State | Published - 1 Feb 2016 |
Bibliographical note
Publisher Copyright:© 2015 .
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics