Graphical sequences and plane trees

TL;DR

The paper addresses the asymptotics of unordered graphical degree sequences by linking them to plane trees embedded in the plane. It employs a Lévy–Khintchine renewal framework that connects a random-walk representation of degree sequences to Walkup's plane-tree counts ${\mathcal{T}}_n$, yielding a precise constant $C = e^{\omega}/\Gamma(1/4)$ with $\omega = \sum_{k\ge1}{\mathcal{T}}_k/(k4^k)$ and the relation $\rho = 1 - e^{-2\omega}$. A central combinatorial bridge is established via a bijection that identifies the Lévy–Khintchine transform of graphical bridges with $2{\mathcal{T}}_n$, and two key lattice-path lemmas connect ${\mathcal{T}}_n$ to areas under lattice paths and to diamond areas of bridges. The results unify infinite-divisibility, renewal theory, and enumerative combinatorics to derive the asymptotics and illuminate the connection between degree sequences and plane-tree structures, with potential applicability to other combinatorial sequences via the Lévy–Khintchine framework.

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 Lévy-Khintchine formula. Our main result complements a result of Stanley, that ordered graphical sequences are related to quasi-forests.

Figures (6)

• Figure 1: The ${\mathcal{T}}_3=4$ rooted unlabeled plane trees with 3 edges.
• Figure 2: A bridge $B$ with diamond area $\sigma(B)=2$, and its lazy version $\Lambda(B)$ below.
• Figure 3: There are $\mathcal{B}_5=38$ graphical bridges of length $10$. Of these, 32 stay within the string of grey diamonds centered along the $x$-axis. The other 6 are depicted above. The top two are irreducible. All others have two irreducible parts. Irreducible parts are separated by solid dots.
• Figure 4: The bijection in \ref{['L_comb2']}.
• Figure 5: Calculating areas in \ref{['L_comb2']}.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Lemma 3: BDK24
• Definition 4
• Definition 5
• Definition 6
• Proposition 7
• proof : Proof of \ref{['T_main2']}
• Definition 8
• Lemma 9