The Leaf Function for Graphs Associated with Penrose Tilings

Abstract

In graph theory, the question of fully leafed induced subtrees has recently been investigated by Blondin Masse et al. in regular tilings of the Euclidian plane and 3-dimensional space. The function L_G that gives the maximum number of leaves of an induced subtree of a graph G of order n, for any n∊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. In particular, we show that 2φn/4φ+1) ≤L_kd(n) ≤ ⌊n/2⌋ + 1, for any n ∊ N, where φ is the golden ratio and L_kd is the leaf function for kites and darts Penrose tilings. As a byproduct, a purely discrete representation of points in the tiling, using quadruples, is described.