Abstract
Using the symmetric form of the Lovász Local Lemma, one can conclude that a k-uniform hypergraph \(\mathcal{H}\) admits a proper 2-colouring if the maximum degree (denoted by Δ) of \(\mathcal{H}\) is at most \(\frac{2^k}{8k}\) independently of the size of the hypergraph. However, this argument does not give us an algorithm to find a proper 2-colouring of such hypergraphs. We call a hypergraph linear if no two hyperedges have more than one vertex in common.
In this paper, we present a deterministic polynomial time algorithm for 2-colouring every k-uniform linear hypergraph with \(\Delta \le 2^{k-k^{\epsilon}}\), where 1/2 < ε< 1 is any arbitrary constant and k is larger than a certain constant that depends on ε. The previous best algorithm for 2-colouring linear hypergraphs is due to Beck and Lodha [4]. They showed that for every δ> 0 there exists a c > 0 such that every linear hypergraph with Δ ≤ 2k − δk and \(k > c\log\log(|E(\mathcal{H})|)\), can be properly 2-coloured deterministically in polynomial time.
Research of the first author is supported by a NSERC graduate scholarship and research grants of Prof. D. Thérien, the second author is supported by a Canada Research Chair in graph theory. We would like to thank an anonymous referee for pointing out the reference [10].
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Chattopadhyay, A., Reed, B.A. (2007). Properly 2-Colouring Linear Hypergraphs. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2007 2007. Lecture Notes in Computer Science, vol 4627. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74208-1_29
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DOI: https://doi.org/10.1007/978-3-540-74208-1_29
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