On the Maximum Cut in Sparse Random Hypergraphs

Abstract

The paper deals with the max-cut problem for random hypergraphs. We consider a binomial model of a random k-uniform hypergraph H(n, k, p) for some fixed $k \ge 3$, growing n and $p=p(n)$. For given natural number q, the max-q-cut for a hypergraph is the maximal possible number of edges that can be properly colored with q colors under the same coloring. Generalizing the known results for graphs we show that in the sparse case (when $p = cn/\left( {\begin{array}{c}n\\ k\end{array}}\right)$ for some fixed $c > 0$ not depending on n) there exists a limit constant $\gamma (c, k, q)$ such that

$$ {\text {max-q-cut}(H(n, k,p)) \over n} \overset{\mathsf {P}}{\longrightarrow } \gamma (c, k, q) $$

as $n \rightarrow +\infty$. We also prove some estimates for $\gamma (c, k, q)$ of the form $A_{k, q} \cdot c + B_{k, q} \cdot \sqrt{c} + o(\sqrt{c})$.