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On the strong chromatic number of a random 3-uniform hypergraph
Discrete Mathematics ( IF 0.770 ) Pub Date : 2020-11-20 , DOI: 10.1016/j.disc.2020.112231
Arseniy E. Balobanov; Dmitry A. Shabanov

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, pn3=cn, 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<rlnr3518lnr13r16, the random hypergraph H(n,3,p) is strongly r-colorable with high probability and, vice versa, for c>rlnr3518lnr+O(lnrr), it is not strongly r-colorable with high probability.

更新日期:2020-11-21
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