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Colouring Arcwise Connected Sets in the Plane II

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Abstract.

Let ? be the family of finite collections ? where ? is a collection of bounded, arcwise connected sets in ℝ2 where for any S,T∈? such that ST≠?, it holds that ST is arcwise connected. Given ? is triangle-free, and provided the chromatic number χ(G) of the intersection graph G=G(?) of ? is sufficiently large, there exists α>1 independent of ? such that there is a subcollection ?⊂? of at most 5 sets with the property that the sets of ? surrounded by ? induce an intersection graph H where .

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Received: November 13, 1995 Final version received: December 3, 1998

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McGuinness, S. Colouring Arcwise Connected Sets in the Plane II. Graphs Comb 17, 135–148 (2001). https://doi.org/10.1007/PL00007235

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  • DOI: https://doi.org/10.1007/PL00007235

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