The maximum degree of planar graphs I. Series-parallel graphs

TL;DR

This paper establishes that the maximum degree of random series-parallel and outerplanar graphs grows proportionally to the logarithm of the number of vertices, with a specific constant factor, in probability and expectation.

Contribution

It proves the asymptotic behavior of the maximum degree in random series-parallel and outerplanar graphs, providing a precise growth rate and constant.

Findings

Maximum degree scales as c log n in probability.

Expected maximum degree is approximately c log n.

Results apply to both series-parallel and outerplanar graphs.

Abstract

We prove that the maximum degree $\Delta_n$ of a random series-parallel graph with $n$ vertices satisfies $\Delta_n/\log n \to c$ in probability, and $\mathbb{E}\, \Delta_n \sim c \log n$ for a computable constant $c>0$. The same result holds for outerplanar graphs.