The interval number of a planar graph is at most three

Abstract : 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. \textit{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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https://hal.archives-ouvertes.fr/hal-02068628
Contributor : Kolja Knauer <>
Submitted on : Friday, March 15, 2019 - 10:19:21 AM
Last modification on : Saturday, August 24, 2019 - 1:07:29 AM

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  • HAL Id : hal-02068628, version 1
  • ARXIV : 1805.02947

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Guillaume Guegan, Kolja Knauer, Jonathan Rollin, Torsten Ueckerdt. The interval number of a planar graph is at most three. 2019. ⟨hal-02068628⟩

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