单位四边形图的几个着色问题
Some colouring problems for unit-quadrance graphs
摘要：
两点$A_1 = (x_1, y_1)$和$A_2 = (x_2, y_2)$之间的四边形是数字$Q (A_1, A_2) = (x_1 - x_2)^2 + (y_1 - y_2)^2$。设$q$为奇素数幂，$F_q$为含$q$元的有限域.单位四边形图$D_q$具有顶点集$F_q^2$，$X, Y \in F_q^2$相邻当且仅当$Q (A_1, A_2) = 1$。本文研究了单位四边形图$D_q$的几个着色问题，并讨论了一些公开问题。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
