Colouring Arcwise Connected Sets in the Plane I

Abstract

Let G be the family of ®nite collections S where S is a collection of bounded, arcwise connected sets in R 2 which for any SY T e S where S T H h, it holds that S T is arcwise connected. We investigate the problem of bounding the chromatic number of the intersection graph G of a collection S e G.Assuming G is triangle-free, suppose there exists a closed Jordan curve C r R 2 such that C intersects all sets of S and for all S e S, the following holds:Here intC and extC denote the regions in the interior, resp. exterior, of C. Such being the case, we shall show that wS is bounded by a constant independent of S.

New Bounds on the Maximum Number of Edges in k-Quasi-Planar Graphs

“…Our proof of Theorem 2 is based on the following result, proved in [10] in a more general setting, for simple K k -free families of compact arc-connected sets in the plane whose intersections with a line are non-empty segments. Special cases of Theorem 4 have been proved by McGuinness [11] for k = 3 and by Suk [18] for y-monotone curves and any k. We will also use the following graph-theoretical result.…”

Triangle-Free Geometric Intersection Graphs with No Large Independent Sets

“…On the other hand, it follows from the result of McGuinness [4] that every triangle-free intersection graph of n segments has chromatic number O(log n) and maximum independent set size (n/ log n).…”

Colouring Arcwise Connected Sets in the Plane II

“…In [3], we de®ned a dendrite to be an arcwise connected set which is a ®nite union of arcs, no subcollection of which contains a closed Jordan curve. According to [Proposition 2.1,3], it suces to consider to prove Theorem 1.1 for collections of dendrites.…”

“…Any of the notation or concepts not explicity de®ned here can be found in the ®rst paper in this sequel (see [3]). For a collection F of subsets of R n we de®ne the intersection graph GF of F to be the graph whose vertices correspond to sets in F where 2 vertices are adjacent if and only if their corresponding sets have nonempty intersection.…”

“…1. As in [3], we de®ne a relation + ( a on 2 collections S; T P G 3 , where T & S, in such a way that T + ( a S D T ! 1 a vS:…”