The odd-distance graph (Q2837316)

The odd-distance graph is defined on the real plane, where two vertices are connected if they are at an odd distance from each other. A related coloring problem is to color the points of the plane so that those at odd distance receive different colors.NEWLINENEWLINENEWLINEIt has been shown that it takes at least four, and at most seven, colors to color points one unit apart in the plane. Such a pair of bounds has not been improved since the 1950s. Regarding the chromatic number for more general odd-distance graphs, although a lower bound of five was shown in [\textit{J. Ardal} et al., Discrete Comput. Geom. 42, No. 2, 132--141 (2009; Zbl 1200.05059)], a finite upper bound does not exist, as was proved, e.g., in [\textit{J. Steinhardt}, Electron. J. Comb. 16, No. 1, Research Paper N12, 7 p. (2009; Zbl 1197.05060)].NEWLINENEWLINENEWLINEThis note describes the current research status of this area, and poses a collection of several related research problems. On the other hand, a more detailed discussion of relevant topics, such as a comparison of the unit-distance graphs and the odd-distance ones and an analysis of a triangular unit grid, can be found in [Ardal, loc. cit.]. Drafts of both [Zbl 1200.05059; Zbl 1197.05060] are available online.