Elsevier

Discrete Mathematics

Volume 122, Issues 1–3, 15 November 1993, Pages 269-286
Discrete Mathematics

Perfect colorings of multipatterns in the plane

https://doi.org/10.1016/0012-365X(93)90302-AGet rights and content
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Abstract

In this paper we consider the problem of finding nontrivial perfect colorings of planar multipatterns. By a ‘multipattern’ we mean a symmetric planar figure, with symmetry group G, having several classes of motifs each transitive under the action of G. A perfect or symmetric coloring, then, is an assignment of colors to the motifs such that each symmetry operation of the group induces a unique permutation of the colors; we will here assume further that the action of G on the colors is transitive. If G is a wallpaper or frieze group, this is always possible and there is a number N(G), the coloring number of G, which is the minimum number (>1) of colors which suffice to color any multipattern with symmetry group G. In the case of finite groups, all multipatterns have nontrivial perfect colorings with the following exceptions: there are no nontrivial perfect colorings if some motif is fixed by all gϵG, or if G is the dihedral group dn with n=2r,r⩾1 and motifs lie on all reflection axes.

For wallpaper groups, 2⩽N(G)⩽25. As a consequence it follows that every periodic k-isohedral tiling of the plane has a perfect coloring using at most 25 colors.

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