On sum coloring and sum multi-coloring for restricted families of graphs

We consider the sum coloring (chromatic sum) and sum multi-coloring problems for restricted families of graphs. In particular, we consider the graph classes of proper intersection graphs of axis-parallel rectangles, proper interval graphs, and unit disk graphs. All the above mentioned graph classes belong to a more general graph class of (k+1)clawfree graphs (respectively, for k = 4, 2, 5). We prove that sum coloring is NP-hard for penny graphs and unit square graphs which implies NP-hardness for unit disk graphs and proper intersection graphs of axis-parallel rectangles. We show a 2-approximation algorithm for unit square graphs, with the assumption that the geometric representation of the graph is given. For sum multi-coloring, we confirm that the greedy first-fit coloring, after ordering vertices by their demands, achieves a k-approximation for the preemptive version of sum multi-coloring on (k + 1)-clawfree graphs. Finally, we study priority algorithms as a model for greedy algorithms for the sum coloring and sum multi-coloring problems. We show various inapproximation results under several natural input representations.