A Triangle-free, 4-chromatic [formula omitted] Euclidean distance graph Scavenger hunt!

Abstract

For d>0, define G(Q3,d) to be the graph whose set of vertices is the rational space Q3, where two vertices are adjacent if and only if they are a Euclidean distance d apart. Let χ(Q3,d) be the chromatic number of such a graph or, in other words, the minimum number of colors needed to color the points of Q3 so that no two points 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 χ(Q3,d)=3. Through numerous efforts over the years, χ(Q3,d) has been determined for many values of d, and for all those distances d where χ(Q3,d) has not been exactly pinned down, it is known that χ(Q3,d)∈{3,4}. In our work, we detail several search algorithms we have employed to find 4-chromatic subgraphs of various graphs G(Q3,d) whose chromatic number was previously unknown. Ultimately, we conjecture that no 3-chromatic G(Q3,d) exists. Along the way, we pose a few related questions that we feel are of interest in their own right.