Perfect precise colorings of plane regular tilings

Abstract

A coloring of a planar tiling T is an assignment of a unique color to each tile of T . If G is the symmetry group of T , we say that the coloring is perfect if every element of G induces a permutation on the finite set of colors. On the other hand, if no two tiles of T sharing the same vertex have the same color, then the coloring is said to be precise. In this work, we obtain perfect precise colorings of some families of k-valent semiregular tilings in the plane using k colors.

Perfect precise colorings of plane semiregular tilings

“…The term precise coloring was coined by Rigby in [3] to refer to a coloring of the regular triangular tiling (3 7 ) in the hyperbolic plane in which no two tiles of the same color share a common vertex. This research is a continuation of work on identifying perfect precise colorings of planar tilings in [4,5]. We demonstrate how to obtain perfect precise colorings with k colors of some families of plane semiregular k-valent tilings where k ≤ 6.…”

Perfect precise colorings of plane semiregular tilings

“…More formally, we note that the case where p 1 , p 2 and p 3 are distinct is covered by Lemma 2.2. The case when T is regular, that is, when p 1 ¼ p 2 ¼ p 3 ¼ p for some p, was considered by Santos & Felix (2011). In this case, the symmetry group of T is G ¼ Ãp32 and the precise coloring of T is perfect because it corresponds to the partition fgðJtÞ j g 2 Gg of T , where t is the p-gon stabilized by the pfold rotation QR and J is the subgroup hPRQRP; Q; Ri of index 3 in G. The last case is, without loss of generality, when q :¼ p 1 is distinct from 2p :¼ p 2 ¼ p 3 for some integer p. Here G ¼ Ãpq2 and the precise coloring of T is perfect because it corresponds to the partition fGt 1 g [ fgðJt 2 Þ j g 2 Gg of T , where t 1 is any q-gon, t 2 is the 2p-gon stabilized by the p-fold rotation QR, and J is the subgroup hPRP; Q; Ri of index 2 in G. Fig.…”

“…This paper is a continuation of work on identifying perfect precise colorings of planar tilings T (Rigby, 1997;Crowe, 1999;Yaz, 2008a;Santos & Felix, 2011). In Section 2, we state general results on perfect precise colorings of T .…”

JPD-Coloring of the Monohedral Tiling for the Plane

“…The notions of tiling by regular polygons in the plane are introduced by Grünbaum and Shephard in [2]. For more details see [3][4][5].…”