Tight Bounds on the Threshold for Permuted k-Colorability

If each edge (u, v) of a graph G = (V,E) is decorated with a permutation πu,v of k objects, we say that it has a permuted k-coloring if there is a coloring σ : V → {1, . . . , k} such that σ(v) 6= πu,v(σ(u)) for all (u, v) ∈ E. Based on arguments from statistical physics, we conjecture that the threshold dk for permuted k-colorability in random graphs G(n,m = dn/2), where the permutations on the edges are uniformly random, is equal to the threshold for standard graph k-colorability. The additional symmetry provided by random permutations makes it easier to prove bounds on dk. By applying the second moment method with these additional symmetries, and applying the first moment method to a random variable that depends on the number of available colors at each vertex, we bound the threshold within an additive constant. Specifically, we show that for any constant ε > 0, for sufficiently large k we have 2k ln k − ln k − 2− ε ≤ dk ≤ 2k ln k − ln k − 1 + ε . In contrast, the best known bounds on dk for standard k-colorability leave an additive gap of about ln k between the upper and lower bounds.