Online Coloring and $L(2,1)$-Labeling of Unit Disk Intersection Graphs

Abstract

In this paper we give a family of online algorithms for the classical coloring and the $L(2,1)$-labeling problems of unit disk intersection graphs. In the $L(2,1)$-labeling we ask for an assignment of nonnegative integers to the vertices of the input graph, such that adjacent vertices get labels that differ by at least 2, and vertices with a common neighbor get different labels. In particular, we present a coloring algorithm with competitive ratio less than 5, which makes it the currently best online coloring algorithm for unit disk intersection graphs. Our algorithms make use of a geometric representation of such graphs and are inspired by previous results but have better competitive ratios. The improvement comes from a novel application of a fractional and a $b$-fold coloring of the plane, which is in turn a variation of the Hadwiger--Nelson problem. Our method can also be adapted successfully for other classes of geometric intersection graphs.