Some colouring problems for unit-quadrance graphs

Abstract

The quadrance between two points $A_1 = (x_1, y_1)$ and $A_2 = (x_2, y_2)$ is the number $Q (A_1, A_2) = (x_1 - x_2)^2 + (y_1 - y_2)^2$. Let $q$ be an odd prime power and $F_q$ be the finite field with $q$ elements. The unit-quadrance graph $D_q$ has the vertex set $F_q^2$, and $X, Y \in F_q^2$ are adjacent if and only if $Q (A_1, A_2) = 1$. In this paper, we study some colouring problems for the unit-quadrance graph $D_q$ and discuss some open problems.