置换k-可染性阈值的紧界
Tight bounds on the threshold for permuted k-colorability
摘要：
如果图G =（V，E）的每条边（u，v）都用k个对象的置换pi_{u,v}来修饰，则称它具有置换k-染色，如果存在从V到{1，.，k}，使得对于E中的所有（u，v），sigma（v）不同于pi_{u,v}（sigma（u））。本文利用统计物理学的观点，对边的排列是均匀随机的随机图G（n，m = dn/2），猜想其排列k-可染性的阈值d_k等于标准图的k-可染性的阈值。随机排列提供的额外对称性使得证明d_k的界限变得更容易。通过将二阶矩方法应用于这些额外的对称性，并将一阶矩方法应用于取决于每个顶点处可用颜色数量的随机变量，我们将阈值限制在一个附加常数内。具体地说，我们证明了对于任何常数k> 0，对于足够大的k，我们有2klnk-lnk-2-k <d_k <2klnk-lnk-1 + k。与此相反，标准k-可染性的d_k上的最佳已知界限在上界和下界之间留下约ln k的加性间隙。
英文摘要：
If each edge (u,v) of a graph G=(V,E) is decorated with a permutation pi_{u,v} of k objects, we say that it has a permuted k-coloring if there is a coloring sigma from V to {1,...,k} such that sigma(v) is different from pi_{u,v}(sigma(u)) for all (u,v) in 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 epsilon > 0, for sufficiently large k we have 2 k ln k - ln k - 2 - epsilon < d_k < 2 k ln k - ln k - 1 + epsilon. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
