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- W4383336654 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." @default.
- W4383336654 created "2023-07-07" @default.
- W4383336654 creator A5025496786 @default.
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- W4383336654 date "2023-11-01" @default.
- W4383336654 modified "2023-09-27" @default.
- W4383336654 title "A Triangle-free, 4-chromatic <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline id=d1e57 altimg=si17.svg><mml:msup><mml:mrow><mml:mi mathvariant=double-struck>Q</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math> Euclidean distance graph Scavenger hunt!" @default.
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- W4383336654 doi "https://doi.org/10.1016/j.dam.2023.06.027" @default.
- W4383336654 hasPublicationYear "2023" @default.
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