Streaming algorithms for 2-coloring uniform hypergraphs
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
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Let H be a k-uniform hypergraph with n vertices. A strong r-coloring is a partition of the vertices into r parts such that each edge of H intersects each part. A strong r-coloring is called equitable if the size of each part is $\lceil n/r \rceil$ or $\lfloor n/r \rfloor$. We prove that for all $a \geq 1$, if the maximum degree of H satisfies $\Delta(H) \leq k^a$, then H has an equitable coloring with $\frac{k}{a \ln k}(1-o_k(1))$ parts. In particular, every k-uniform hypergraph with maximum degree O(k) has an equitable coloring with $\frac{k}{\ln k}(1-o_k(1))$ parts. The result is asymptotically tight. The proof uses a double application of the nonsymmetric version of the Lovász local lemma.