The Leaf Function of Penrose P2 Graphs

TL;DR

The paper studies the leaf function $L_{P2}$ for Penrose $P2$-graphs (duals of kite-and-dart tilings) and the problem of identifying fully leafed induced subtrees. It proves an explicit upper bound on $L_{P2}(n)$ and constructs an infinite family of fully leafed caterpillars $(C_n)$ that achieve this bound, using a finite poset of $3$-internal-regular subtrees and a $\varphi^3$-decomposition tied to Star tilings. A closed form for $L_{P2}(n)$ and its asymptotics $L_{P2}(n)\sim 8n/17$ are derived, with a practical procedure to generate large optimal subtrees. The work connects Penrose tilings, local isomorphism, Ammann bars, and Star tilings to provide constructive realizations and opens questions about saturation and extensions to other Penrose types and MLD classes.

Abstract

We study a graph-theoretic problem in the Penrose P2-graphs which are the dual graphs of Penrose tilings by kites and darts. Using substitutions, local isomorphism and other properties of Penrose tilings, we construct a family of arbitrarily large induced subtrees of Penrose graphs with the largest possible number of leaves for a given number $n$ of vertices. These subtrees are called fully leafed induced subtrees. We denote their number of leaves $L_{P2}(n)$ for any non-negative integer $n$, and the sequence $\left(L_{P2}(n)\right)_{n\in\mathbb{N}}$ is called the leaf function of Penrose P2-graphs. We present exact and recursive formulae for $L_{P2}(n)$, as well as an infinite sequence of fully leafed induced subtrees, which are caterpillar graphs. In particular, our proof relies on the construction of a finite graded poset of 3-internal-regular subtrees.

Figures (19)

• Figure 1: Fully leafed induced subtrees, (a) in a finite graph, (b) in the triangular lattice, (c) in the hexagonal lattice, (d) in the square lattice, and (e) in the cubic lattice. Vertices are colored according to their degree in the induced subtree (except for the finite graph). Subtrees (a) and (b) are caterpillars, subtrees (c), (d) and (e) are not.
• Figure 2: An excerpt of a P2 tiling, also called kites and darts tiling, with its adjacency graph superimposed. The kites and darts are respectively colored yellow and orange. The graph vertices are depicted with blue dots and the edges in cyan.
• Figure 3: The kite and dart, followed by the seven vertex configurations in a Penrose P2 tiling. The colored vertices on the kite and dart on the left give the assembly rules: the colors have to match as one assembles the tiles, so that there is either a black or a white circle at each vertex. No other vertex configuration than those above is allowed -- in particular, no rhombus can be formed. All patches are considered up to isometry.
• Figure 4: The seven kingdoms of kites and darts tilings.
• Figure 5: Each black point corresponds to an occurrence of a King, with the same kingdom around. Two occurrences of the King's kingdom are colored in yellow and orange, in different orientations and separated by a (finite) Conway worm colored in blue, as well as the other finite worms surrounding the highlighted kingdoms. Each worm could be flipped around a central axis parallel to the Ammann bar (red line) crossing the worm -- the bar being flipped around the same axis in that case, for instance touching the kingdom on the left instead of that on the right.

Theorems & Definitions (21)

• Theorem 1
• Proposition 1
• Lemma 1: PB2020
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof