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Authors:	James H Schmerl

Institution:	Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009

Abstract:	If $1 \leq m \leq n$ and $A \subseteq {\mathbb R}$ , then define the graph to be the graph whose vertex set is ${\mathbb R}^n$ with two vertices $x,y \in {\mathbb R}^n$ being adjacent iff there are distinct $u,v \in A^m$ such that $\Vert x-y\Vert = \Vert u-v\Vert$ . For various and and various , typically $A = {\mathbb Q}$ or $A = {\mathbb Z}$ , the graph can be properly colored with $\omega$ colors. It is shown that in some cases such a coloring $\varphi : {\mathbb R}^n \longrightarrow\omega$ can also have the additional property that if $\alpha : {\mathbb R}^m \longrightarrow{\mathbb R}^n$ is an isometric embedding, then the restriction of $\varphi$ to $\alpha(A^m)$ is a bijection onto $\omega$ .

Keywords:	Graph coloring distance graphs Steinhaus property

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