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Abstract

Let $G$ be a graph and $e_1,\cdots ,e_n$ be $n$ distinct vertices. Let $\rho$ be the metric on $G$. The code map on vertices, corresponding to this list, is $c(x)=(\rho (x,e_1),\cdots ,\rho (x,e_n))$. This paper introduces a variation: begin with $V\subseteq\bbz^n$ for some $n$, and consider assignments of edges $E$ such that the identity function on $V$ is a code map for $G=(V,E)$. Refer to such a set $E$ as a {\em code-match.}

An earlier paper classified subsets of $V$ for which at least one code-match exists. We prove \begin{itemize} \item If there is a code-match $E$ for which $(V,E)$ is bipartite, than $(V,E)$ is bipartite for every code-match $E$. \item If there is a code-match $E$ for which $(V,E)$ is a tree, then $E$ is unique. \item There exists a code-match $E$ such that $(V,E)$ has a $(2^{n-1}+1)$-vertex-coloring. \end{itemize}

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This work is licensed under a Creative Commons Attribution 4.0 License.

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