The Leaf Function for Graphs Associated with Penrose Tilings

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 LP 2(n) for any non-negative integer n, and the sequence (LP 2(n)) n∈N is called the leaf function of Penrose P2-graphs. We present exact and recursive formulae for LP 2(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.

HBS Tilings Extended: State of the Art and Novel Observations

“…It was derived from P2 tilings, but not in the same way as in Figure 4b. The construction is still quite simple and in addition to the ϕ 2 -composition mentioned above, we have a ϕ-composition, which is what interested us for the construction in [10]. Vertex colors are strongly related to vertex configurations in P2, hence to vertex configurations in HBS tilings when you look at the compositions.…”

“…As we found it difficult to know whether some of our findings had already been published and if so, including which information, it seemed a state of the art would be useful to us and might be for others too. But most of all, we have new findings which we rely on for an article to come [10].…”

The Leaf Function of Graphs Associated with Penrose Tilings

“…Note that Sections 1-2 and 4-6 of this article were already in an earlier version of the paper, originally presented at IEEE Graph Computing 2019 [15]. In this extended version, we provide more details about the computational aspects in Sections 3, 6 and 7.…”