The Chromatic Number of the Disjointness Graph of the Double Chain

TL;DR

This paper determines the exact chromatic number of the disjointness graph for a specific point configuration called the double chain, providing a precise formula based on the number of points in the chains.

Contribution

It establishes the exact chromatic number for the disjointness graph of the double chain configuration, a problem previously unresolved.

Findings

The chromatic number equals k + l - floor(sqrt(2l + 1/4) - 1/2).

The derived formula is both necessary and sufficient.

The result advances understanding of geometric graph coloring in specific configurations.

Abstract

Let $P$ be a set of $n\geq 4$ 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 $l \ge k$ points in the lower convex chain, then $k+l- \left\lfloor \sqrt{2l+\frac{1}{4}} - \frac{1}{2}\right\rfloor$ colors are needed and that this number is sufficient.