Fractional and j-Fold Coloring of the Plane

Jarosław Grytczuk, Konstanty Junosza-Szaniawski, Joanna Sokół, Krzysztof Węsek
2016 Discrete & Computational Geometry  
We present results referring to the Hadwiger-Nelson problem which asks for the minimum number of colours needed to colour the plane with no two points at distance 1 having the same colour. Exoo considered a more general problem concerning graphs G_[a,b] with R^2 as the vertex set and two vertices adjacent if their distance is in the interval [a,b]. Exoo conjectured χ(G_[a,b]) = 7 for sufficiently small but positive difference between a and b. We partially answer this conjecture by proving that
more » ... (G_[a,b]) ≥ 5 for b > a. A j-fold colouring of graph G = (V,E) is an assignment of j-elemental sets of colours to the vertices of G, in such a way that the sets assigned to any two adjacent vertices are disjoint. The fractional chromatic number χ_f(G) is the infimum of fractions k/j for j-fold colouring of G using k colours. We generalize a method by Hochberg and O'Donnel (who proved that G_[1,1]≤ 4.36) for fractional colouring of graphs G_[a,b], obtaining a bound dependant on 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 colouring 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.
doi:10.1007/s00454-016-9769-3 fatcat:aetxp63a2raaffvf77oywtiodm