Cayley trees and increasing 1,2-trees: let's twist!

TL;DR

This work studies increasing $(\le 2)$-trees, a labeled class of chordal graphs with treewidth at most 2, and proves they are equinumerous with Cayley trees, with the count for size $n$ being $n^{n-2}$ and the number of triangles corresponding to the number of twists in Cayley trees. It presents three independent proofs: a recursive-decomposition argument paralleling Shor’s approach for Cayley trees, a generating-function PDE solution that expresses the counting function in terms of Cayley’s generating function, and an explicit bijection $\tau$ that preserves the relevant statistics via an intermediate plane forest. The paper further provides linear-time uniform random samplers for both families by leveraging the bijection and structural decompositions. Overall, the results illuminate a deep combinatorial link between increasing 1,2-trees and Cayley trees, enabling efficient generation and potential generalizations to higher treewidth regimes $(\le k)$ for $k\ge3$.

Abstract

An increasing 1,2-tree is a labeled graph formed by starting with a vertex and then repeatedly attaching a leaf to a vertex or a triangle to an edge, the labeling of the vertices corresponding to the order in which the vertices are added. Equivalently, increasing 1,2-trees are connected chordal graphs of treewidth at most 2 labeled with a reversed perfect elimination ordering. We prove that this family is equinumerous with Cayley trees, which are unconstrained labeled trees. In particular, the number of triangles in an increasing 1,2-tree corresponds to the number of twists. A twist (also called improper edge) is an edge whose endpoint closer to vertex 1 has a greater label than some vertex in the subtree rooted at the other endpoint of the edge. We provide three proofs of this result, the rst being based on similar recursive decompositions, the second on the resolution of generating functions, and the third describing a bijection. Finally, we propose ecient random generators for these two combinatorial families.

Figures (14)

• Figure 1: The 16 increasing 1,2-trees with $4$ vertices, gathered by their numbers of edges.
• Figure 2: The 16 Cayley trees with $4$ vertices. The trees are sorted by the number of twists ($0$ twist in the first row, $1$ twist in the second row, $2$ twists in the bottom row). The twists are represented differently (as if the edges have really been twisted) than the increasing edges.
• Figure 3: Shor's idea to enumerate all Cayley trees while controlling the number of twists
• Figure 4: The first three levels of the generating tree of increasing 1,2-trees (left) and the generating tree of Cayley trees (right).
• Figure 5: Overview of the bijection from increasing 1,2-trees to Cayley trees.

Theorems & Definitions (41)

• Definition 1: Increasing $(\leqslant k)$-trees
• Definition 2: Cayley trees
• Definition 3: increasing edge, twist
• Definition 4: $\min(S)$
• Theorem 5
• Corollary 6
• Proposition 7
• proof
• Proposition 8
• proof