Coloring Graph Powers: Graph Product Bounds and Hardness of Approximation

We consider the question of computing the strong edge coloring, square graph coloring, and their generalization to coloring the k power of graphs. These problems have long been studied in discrete mathematics, and their “chaotic” behavior makes them interesting from an approximation algorithm perspective: For k = 1, it is well-known that vertex coloring is “hard” and edge coloring is “easy” in the sense that the former has an n1− hardness while the latter admits a (4/3)-approximation algorithm. However, vertex coloring becomes easier (can be O( √ n)-approximated) for k = 2 while edge coloring seems to become much harder (no known O(n1− ) approximation algorithm) for k ≥ 2. In this paper, we show several new hardness of approximation results that clarify the approximability landscape of these problems. First, we confirm that edge coloring indeed becomes computationally harder when k > 1: we prove a hardness of n1/3− for k ∈ {2, 3} and n1/2− for k ≥ 4 (previously, only NP-hardness for k = 2 is known). Moreover, for vertex coloring, we prove a hardness of n1/2− for all k, which is tight for all even k. We also prove hardness of maximum clique and stable set (a.k.a. independent set) problems on graph powers. These results rely on a common simple technique of proving bounds via fractional coloring. This technique also allows us to prove some new bounds on graph products. In addition, we include our proof of Erdös and Nešetřil conjecture on cographs using a charging scheme technique.