Additive K-colorable extensions of the rational plane

✍ Scribed by Klaus G. Fischer

• Publisher: Elsevier Science
• Year: 1990
• Tongue: English
• Weight: 947 KB
• Volume: 82
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(90)90324-b

✦ Synopsis

Let F be a field, Q c F c 5S and consider Fd as a graph with vertices the points of Fd and an edge between two points if their Euclidean distance is 1. Let C,, (Fd) be the subgroup of Fd generated by the unit vectors $. If G is a group of order k, then a group homomorphism ~1: &, (Fd)-+ G for which u(E) # 0 whenever II511 = 1 is said to be an additive k-coloring of Fd. The known 2 and 4-colorings of Q3 and @ respectively, are shown to be additive. If N is a square free integer, then it is shown that Q(<?$* h as N # 2 mod 3, then a;P(V%)* an additive 2-coloring iff N # 3 mod 4. If h as an additive 3-coloring. Hence, it follows that the chromatic number of Q(fi)' is 3. The existence of additive colorings on Q(V??)' for the remaining cases is also discussed. Additive k-colorings constrain cycles in Fd to satisfy group identities. Hence, it is shown for example, that if F2 is 2-colorable and if ti $ F, then FZ contains no regular polygon except for the square. This generalizes the classical result known for the rational plane.