# Fractional and j-Fold Coloring of the Plane

Discrete & Computational Geometry, Mar 2016

We present results referring to the Hadwiger–Nelson problem which asks for the minimum number of colors needed to color the plane with no two points at distance 1 having the same color. Exoo considered a more general problem concerning graphs $G_{[a,b]}$ with $\mathbb {R}^2$ as the vertex set and two vertices adjacent if their distance is in the interval [a, b]. Exoo conjectured $\chi (G_{[a,b]}) = 7$ for sufficiently small but positive difference between a and b. We partially answer this conjecture by proving that $\chi (G_{[a,b]}) \geqslant 5$ for $b > a$. A j-fold coloring of a graph $G = (V,E)$ is an assignment of j-elemental sets of colors to the vertices of G, in such a way that the sets assigned to any two adjacent vertices are disjoint. The fractional chromatic number $\chi _f(G)$ is the infimum of fractions k / j for j-fold coloring of G using k colors. We generalize a method by Hochberg and O’Donnel (who proved that $G_{[1,1]} \leqslant 4.36$) for the fractional coloring of graphs $G_{[a,b]}$, obtaining a bound dependent on $\frac{a}{b}$. We also present few specific and two general methods for j-fold coloring of $G_{[a,b]}$ for small j, in particular for $G_{[1,1]}$ and $G_{[1,2]}$. The j-fold coloring for small j has strong practical motivation especially in scheduling theory, while graph $G_{[1,2]}$ is often used to model hidden conflicts in radio networks.

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Jarosław Grytczuk, Konstanty Junosza-Szaniawski, Joanna Sokół, Krzysztof Węsek. Fractional and j-Fold Coloring of the Plane, Discrete & Computational Geometry, 2016, 594-609, DOI: 10.1007/s00454-016-9769-3