The interval number of a planar graph is at most three
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The interval number of a graph G is the minimum k such that one can assign to each vertex of G a union of k intervals on the real line, such that G is the intersection graph of these sets, i.e., two vertices are adjacent in G if and only if the corresponding sets of intervals have non-empty intersection. In 1983 Scheinerman and West [The interval number of a planar graph: Three intervals suffice. J. Comb. Theory, Ser. B, 35:224--239, 1983] proved that the interval number of any planar graph is at most 3. In this paper we present a flaw in the original proof and give another, different, and shorter proof of this result.
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