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Zbl 0744.05061
Fischer, Klaus G.
Additive $k$-colorable extensions of the rational plane. (English)
[J] Discrete Math. 82, No.2, 181-195 (1990). ISSN 0012-365X

Summary: Let $F$ be a field, $\bbfQ\subset F\subset\bbfR$ and consider $F\sp d$ as a graph with vertices the points of $F\sp d$ and an edge between two points if their Euclidean distance is 1. Let $\Sigma\sb 0(F\sp d)$ be the subgroup of $F\sp d$ generated by the unit vectors $\xi$. If $G$ is a group of order $k$, then a group homomorphism $v: \Sigma\sb 0(F\sp d)\to G$ for which $v(\xi)\ne 0$ whenever $\Vert\xi\Vert=1$ is said to be an additive $k$-coloring of $F\sp d$. The known 2 and 4-colorings of $\bbfQ\sp 3$ and $\bbfQ\sp 4$ respectively, are shown to be additive. If $N$ is a square free integer, then it is shown that $\bbfQ(\sqrt N)\sp 2$ has an additive 2-coloring iff $N\ne 3 \mod 4$. If $N\ne 2 \mod 3$, then $\bbfQ(\sqrt N)\sp 2$ has an additive 3-coloring. Hence, it follows that the chromatic number of $\bbfQ(\sqrt 3)\sp 2$ is 3. The existence of additive colorings on $\bbfQ(\sqrt N)\sp 2$ for the remaining cases is also discussed.\par Additive $k$-colorings constrain cycles in $F\sp d$ to satisfy group identities. Hence, it is shown for example, that if $F\sp 2$ is 2- colorable and if $\sqrt 2\not\in F$, then $F\sp 2$ contains no regular polygon except for the square. This generalizes the classical result known for the rational plane.

MSC 2000:: *05C99 Graph theory
05C15 Chromatic theory of graphs and maps

Keywords: additive $k$-coloring; chromatic number; rational plane

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