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Coloring
Authors:James H. Schmerl
Affiliation: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 $G(A,m,n)$ 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 $Vert x-yVert = Vert u-vVert$. For various $m$ and $n$ and various $A$, typically $A = {mathbb Q}$ or $A = {mathbb Z}$, the graph $G(A,m,n)$ can be properly colored with $omega$ colors. It is shown that in some cases such a coloring $varphi : {mathbb R}^n longrightarrowomega$ 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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