On two-coloring bipartite uniform hypergraphs

Abstract

Of a given bipartite graph $G=(V,E)$, it is elementary to construct a bipartition in time $O(|V|+|E|)$. For a given $k$-graph $H=H^{(k)}$ with $k\ge 3$ fixed, Lovász proved that deciding whether $H$ is bipartite is NP-complete. Let ${\mathcal{B}}_n$ denote the collection of all $[n]$-vertex bipartite $k$-graphs. We construct, of a given $H\in {\mathcal{B}}_n$, a bipartition in time averaging $O(n^k)$ over the class ${\mathcal{B}}_n$. We provide two proofs of our result. When $k=3$, this result expedites one of Person and Schacht.

15 pages