Abstract
Discrete Algorithms In this paper we consider the problem of deciding whether a given r-uniform hypergraph H with minimum vertex degree at least c\binom|V(H)|-1r-1, or minimum degree of a pair of vertices at least c\binom|V(H)|-2r-2, has a vertex 2-coloring. Motivated by an old result of Edwards for graphs, we obtain first optimal dichotomy results for 2-colorings of r-uniform hypergraphs. For each problem, for every r≥q 3 we determine a threshold value depending on r such that the problem is NP-complete for c below the threshold, while for c strictly above the threshold it is polynomial. We provide an algorithm constructing the coloring with time complexity O(n^\lfloor 4/ε\rfloor+2\log n) with some ε>0. This algorithm becomes more efficient in the case of r=3,4,5 due to known Turán numbers of the triangle and the Fano plane. In addition, we determine the computational complexity of strong k-coloring of 3-uniform hypergraphs H with minimum vertex degree at least c\binom|V(H)|-12, for some c, leaving a gap for k≥q 5 which vanishes as k→ ∞.
Highlights
A hypergraph H = (V, E) is a finite set of vertices V together with a family E of distinct, nonempty subsets of vertices called edges
We provide an algorithm constructing the coloring with time complexity O(nC ), for some C = C(c, r) > 0
The property of hypergraph 2-colorability has been studied since the paper of Bernstein [2] and it has got its other name, Property B, after him
Summary
If we disregard the minimum degree condition by setting c = 0, we get the classical problem which asks whether a given r-graph admits a k-coloring. Definition 1.6 For fixed integers r, 1 ≤ l ≤ r − 1 and k ≥ r, and a real number 0 ≤ c ≤ 1, define the problem Πrs,l(k, c) as follows: Output: Is H strong k-colorable (χs(H) ≤ k)? Χs(H) = χ(Gr(H)), the ordinary chromatic number of the clique graph of H Using this relation together with Theorem 1.2 and complementing it with a proof of NP-completeness, we obtain the following result. The paper is concluded with some final remarks and open questions
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