[PDF] A Triangle-free, 4-chromatic $\mathbb{Q}^3$ Euclidean Distance Graph Scavenger Hunt!-论文阅读讨论-ReadPaper

摘要

For $d > 0$, define $G(\mathbb{Q}^3, d)$ to be the graph whose set ofvertices is the rational space $\mathbb{Q}^3$, where two vertices are adjacentif and only if they are a Euclidean distance $d$ apart. Let $\chi(\mathbb{Q}^3,d)$ be the chromatic number of such a graph or, in other words, the minimumnumber of colors needed to color the points of $\mathbb{Q}^3$ so that no twopoints at distance $d$ apart receive the same color. An open problem,originally posed by Benda and Perles in the 1970s, asks if there exists $d$such that $\chi(\mathbb{Q}^3, d) = 3$. Through numerous efforts over the years,$\chi(\mathbb{Q}^3, d)$ has been determined for many values of $d$, and for allthose distances $d$ where $\chi(\mathbb{Q}^3, d)$ has not been exactly pinneddown, it is known that $\chi(\mathbb{Q}^3, d) \in \{3,4\}$. In our work, wedetail several search algorithms we have employed to find $4$-chromaticsubgraphs of various graphs $G(\mathbb{Q}^3, d)$ whose chromatic number waspreviously unknown. Ultimately, we conjecture that no $3$-chromatic$G(\mathbb{Q}^3, d)$ exists. Along the way, we pose a few related questionsthat we feel are of interest in their own right.

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