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 th 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 n 1 − e hardness while the latter admits a \((1+1/\varDelta)\)-approximation algorithm, where \(\varDelta\) is the maximum degree of a graph. However, vertex coloring becomes easier (can be \(O(\sqrt{n})\)-approximated) for k = 2 while edge coloring seems to become much harder (no known O(n 1 − e )-approximation algorithm) for k ≥ 2.