Abstract
If each edge (u,v) of a graph G = (V,E) is decorated with a permutation π u,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) ≠ π u,v (σ(u)) for all (u,v) ∈ E. Based on arguments from statistical physics, we conjecture that the threshold d k 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 d k . 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
In contrast, the best known bounds on d k for standard k-colorability leave an additive gap of about ln k between the upper and lower bounds.
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Dani, V., Moore, C., Olson, A. (2012). Tight Bounds on the Threshold for Permuted k-Colorability. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2012 2012. Lecture Notes in Computer Science, vol 7408. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32512-0_43
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DOI: https://doi.org/10.1007/978-3-642-32512-0_43
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