Abstract This paper deals with estimating the threshold for the strong r -colorability of a random 3-uniform hypergraph in the binomial model H ( n , 3 , p ) . A vertex coloring is said to be strong for a hypergraph if every two vertices sharing a common edge are colored with distinct colors. It is known that the threshold corresponds to the sparse case, when the expected number of edges is a linear function of n , p n 3 = c n , and c > 0 depends on r , but not on n . We establish the threshold as a bound on the parameter c up to an additive constant. In particular, by using the second moment method we prove that for large enough r and c r ln r 3 − 5 18 ln r − 1 3 − r − 1 ∕ 6 , the random hypergraph H ( n , 3 , p ) is strongly r -colorable with high probability and, vice versa, for c > r ln r 3 − 5 18 ln r + O ( ln r ∕ r ) , it is not strongly r -colorable with high probability.

中文翻译：

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关于随机3-均匀超图的强色数

摘要 本文涉及估计二项式模型 H ( n , 3 , p ) 中随机 3-均匀超图的强 r 可着色性的阈值。如果每两个共享公共边的顶点都用不同的颜色着色，则称顶点着色对于超图是强的。众所周知，阈值对应于稀疏情况，当期望的边数是 n 的线性函数时， pn 3 = cn ，并且 c > 0 取决于 r ，但不取决于 n 。我们将阈值建立为参数 c 到一个加性常数的界限。特别地，通过使用二阶矩方法，我们证明对于足够大的 r 和 cr ln r 3 − 5 18 ln r − 1 3 − r − 1 ∕ 6 ，随机超图 H ( n , 3 , p ) 是强 r -以高概率着色，反之亦然，对于 c > r ln r 3 − 5 18 ln r + O ( ln r ∕ r ) ，