The Chromatic Number of the Convex Segment Disjointness Graph

Abstract

Let P be a set of n ≥ 4 points in general position in the plane. Consider all the closed straight line segments with both endpoints in P . Suppose that these segments are colored with the rule that disjoint segments receive different colors. In this paper we show that if P is the point configuration known as the double chain, with k points in the upper convex chain and l ≥ k points in the lower convex chain, then k + l − 2l + 1 4 − 1 2 colors are needed and that this number is sufficient.

Blocking the $$k$$ k -Holes of Point Sets in the Plane

“…Let χ(D n ) denote the chromatic number of D n . A lower bound on this value was obtained by Fabila-Monroy and Wood in [11], while an upper bound was obtained by Dujmović and Wood in [8]. Both bounds combine into the following theorem:…”

“…Then D n has n 2 − n vertices. It is easy to see from the proof of Theorem 2.2 in [11], that the chromatic number of D n satisfies:…”

On the connectivity of the disjointness graph of segments of point sets in general position in the plane

“…The graph on the right is the edge disjointness graph D(P ) corresponding to P . value of χ(D(P )) is known only for two particular cases: when P is in convex position Fabila-Monroy and Wood (2011); Jonsson (2011), and when P is the double chain Fabila-Monroy et al (2020). In 2017 Pach, Tardos, and Tóth Pach et al (2017) studied the chromatic number and the clique number of D(P ) in the more general setting of R d for d ≥ 2, i.e., when P is a subset of R d .…”

An Upper Bound Asymptotically Tight for the Connectivity of the Disjointness Graph of Segments in the Plane

“…The problem of determining the exact value of χ(D(P)) remains open in general. On the other hand, there are only two families of point sets for which the exact value of χ(D(P)) is known: when P is in convex position [11,12], and when P is the double chain [13]. The connectivity κ(D(P)) of D(P) was studied by Leaños, Ndjatchi, and Ríos-Castro in [14], where it was shown that κ(D(P)) ≥…”