On-line coloring between two lines

TL;DR

This paper studies online graph coloring for intersection graphs of convex sets between two lines, providing algorithms with bounded colors for certain classes and exploring the complexity and poset relations of these graphs.

Contribution

It introduces an online coloring algorithm for convex sets between two lines with a cubic bound on colors and analyzes the complexity and poset structure of these graphs.

Findings

The algorithm uses O(ω^3) colors for graphs with maximum clique size ω.

Intersection graphs of segments attached to a single line can require arbitrarily many colors.

The on-line coloring problem for curves between two lines is as hard as the chain partition problem for arbitrary posets.

Abstract

We study on-line colorings of certain graphs given as intersection graphs of objects "between two lines", i.e., there is a pair of horizontal lines such that each object of the representation is a connected set contained in the strip between the lines and touches both. Some of the graph classes admitting such a representation are permutation graphs (segments), interval graphs (axis-aligned rectangles), trapezoid graphs (trapezoids) and cocomparability graphs (simple curves). We present an on-line algorithm coloring graphs given by convex sets between two lines that uses $O(\omega^3)$ colors on graphs with maximum clique size $\omega$. In contrast intersection graphs of segments attached to a single line may force any on-line coloring algorithm to use an arbitrary number of colors even when $\omega=2$. The {\em left-of} relation makes the complement of intersection graphs of objects between two lines into a poset. As an aside we discuss the relation of the class $\mathcal{C}$ of posets obtained from convex sets between two lines with some other classes of posets: all $2$-dimensional posets and all posets of height $2$ are in $\mathcal{C}$ but there is a $3$-dimensional poset of height $3$ that does not belong to $\mathcal{C}$. We also show that the on-line coloring problem for curves between two lines is as hard as the on-line chain partition problem for arbitrary posets.