The genus distribution of cubic graphs and asymptotic number of rooted cubic maps with high genus

Let $C_{n,g}$ be the number of rooted cubic maps with $2n$ vertices on the orientable surface of genus $g$. We show that the sequence $(C_{n,g}:g\ge 0)$ is asymptotically normal with mean and variance asymptotic to $(1/2)(n-\ln n)$ and $(1/4)\ln n$, respectively. We derive an asymptotic expression for $C_{n,g}$ when $(n-2g)/\ln n$ lies in any closed subinterval of $(0,2)$. Using rotation systems and Bender's theorem about generating functions with fast-growing coefficients, we derive simple asymptotic expressions for the numbers of rooted regular maps, disregarding the genus. In particular, we show that the number of rooted cubic maps with $2n$ vertices, disregarding the genus, is asymptotic to $\frac{3}{\pi}\,n!6^n$

Brief introduction to tropical geometry

The paper consists of lecture notes for a mini-course given by the authors at the G\"okova Geometry \& Topology conference in May 2014. We start the exposition with tropical curves in the plane and their applications to problems in classical enumerative geometry, and continue with a look at more general tropical varieties and their homology theories.Comment: 75 pages, 37 figures, many examples and exercise

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement