The Leaf Function of Graphs Associated with Penrose Tilings

Abstract

In graph theory, the question of fully leafed induced subtrees has recently been investigated by Blondin Massé et al in regular tilings of the Euclidian plane and 3-dimensional space. The function LG that gives the maximum number of leaves of an induced subtree of a graph $G$ of order $n$, for any $n\in \N$, is called leaf function. This article is a first attempt at studying this problem in non-regular tilings, more specifically Penrose tilings. We rely not only on geometric properties of Penrose tilings, that allow us to find an upper bound for the leaf function in these tilings, but also on their links to the Fibonacci word, which give us a lower bound. Our approach rely on a purely discrete representation of points in the tilings, thus preventing numerical errors and improving computation efficiency. Finally, we present a procedure to dynamically generate induced subtrees without having to generate the whole patch surrounding them.

The Leaf Function for Graphs Associated with Penrose Tilings

“…Proof. The result is proved in [PBM20], but for sake of completeness, we include it here. It suffices to consider all possible neighborhoods of a kite and a dart in a P2 tiling, shown in Figure 6 (detailing first row of Figure 8 and adding one configuration).…”

“…Joining two adjacent Star centers with a segment, one can see HBS shapes appear: hexagons, boats and stars 3 (Figure 15). HBS tilings are well known but, to our knowledge, this construction is new (see [Por23] for a recent review of the existing literature), and the tiles obtained correspond to the φ 3 -composition of the usual HBS tiles we would obtain by composing kites and darts. In addition to this difference in decorations, by coloring the vertices of the HBS shapes according to the type of flower, the distribution of colors is unique on the vertices of hexagons and boats, while the stars come in three types, with either 0, 1 or 2 red vertices.…”

A General Approach to Ammann Bars for Aperiodic Tilings

“…1) were soon exhibited and aroused interest among physicists but not that much among mathematicians -who wanted less tiles than in Penrose's tilesets. As serendipity works, we rediscovered HBS shapes but with different decorations and forcing rules, while working on a combinatorial optimization problem on graphs defined by kites-and-darts Penrose tilings (type P2), as described in [PBM20]. Indeed, the stars (composed with five darts) are an optimal pattern for us and we were looking at paths that would join stars with the minimum possible distance.…”

Ammann Bars for Octagonal Tilings

“…Generally speaking, we do not know much about Ammann bars and for now each family of aperiodic tilings has to be observed as an example. Yet they can reveal quite useful to study the structure of tilings, and were used by Porrier and Blondin Massé [PBM20] to solve a combinatorial optimization problem on graphs defined by Penrose tilings.…”