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 $H\left(n,3,p\right)$. 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 $n$, $p\left(\genfrac{}{}{0}{}{n}{3}\right)=cn$, 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 $c<\frac{rlnr}{3}-\frac{5}{18}lnr-\frac{1}{3}-{r}^{-1∕6}$, the random hypergraph $H\left(n,3,p\right)$ is strongly $r$-colorable with high probability and, vice versa, for $c>\frac{rlnr}{3}-\frac{5}{18}lnr+O\left(lnr∕r\right)$, it is not strongly $r$-colorable with high probability.

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