Graphical sequences and plane trees

TL;DR

This paper explores the asymptotic enumeration of graphical degree sequences and their connection to plane trees, using advanced probabilistic and combinatorial methods to establish new bijections and formulas.

Contribution

It introduces a novel bijective connection between random walk trajectories and plane trees, linking graphical sequences to additive number theory and Lévy processes.

Findings

Asymptotic count of graphical sequences is proportional to C4^n/n^{3/4}.

Established a bijection between random walks and rooted plane trees.

Connected graphical sequences to Lévy-Khintchine formula and Walkup's enumeration.

Abstract

Balister, the second author, Groenland, Johnston and Scott recently showed that there are asymptotically $C4^n/n^{3/4}$ many unordered sequences that occur as degree sequences of graphs. Combining limit theory for infinitely divisible distributions with a new bijective connection between a class of random walk trajectories and a subset counting formula from additive number theory, we describe $C$ in terms of Walkup's number of rooted plane trees. The bijection is related to an instance of the L\'evy-Khintchine formula. Our main result complements a result of Stanley, that ordered graphical sequences are related to quasi-forests.