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arXiv · 2312.08262

The Leaf Function of Penrose P2 Graphs

Carole Porrier·Alain Goupil·Alexandre Blondin Massé
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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.

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Carole Porrier, Alain Goupil, Alexandre Blondin Massé. 2023-12-13. The Leaf Function of Penrose P2 Graphs. https://doi.org/10.46298/dmtcs.13662

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