The Chromatic Number of Ordered Graphs With Constrained Conflict Graphs

TL;DR

This paper introduces a generalized framework for conflict graphs in ordered graphs using matrices, and studies how these conflicts influence the chromatic number under various constraints.

Contribution

It defines a new matrix-based conflict graph model that unifies many graph concepts and determines maximum chromatic numbers for specific cases of this model.

Findings

Exact values of maximum chromatic numbers for single-row matrices with entries in {-1,0,1}

Analysis of cases with two-row matrices and arbitrary entries

Generalization of concepts like page-number, queue-number, and interval chromatic number

Abstract

An ordered graph $G$ is a graph whose vertex set is a subset of integers. The edges are interpreted as tuples $(u,v)$ with $u < v$. For a positive integer $s$, a matrix $M \in \mathbb{Z}^{s \times 4}$, and a vector $\mathbf{p} = (p,\ldots,p) \in \mathbb{Z}^s$ we build a conflict graph by saying that edges $(u,v)$ and $(x,y)$ are conflicting if $M(u,v,x,y)^\top \geq \mathbf{p}$ or $M(x,y,u,v)^\top \geq \mathbf{p}$, where the comparison is componentwise. This new framework generalizes many natural concepts of ordered and unordered graphs, such as the page-number, queue-number, band-width, interval chromatic number and forbidden ordered matchings. For fixed $M$ and $p$, we investigate how the chromatic number of $G$ depends on the structure of its conflict graph. Specifically, we study the maximum chromatic number $X_\text{cli}(M,p,w)$ of ordered graphs $G$ with no $w$ pairwise conflicting edges and the maximum chromatic number $X_\text{ind}(M,p,a)$ of ordered graphs $G$ with no $a$ pairwise non-conflicting edges. We determine $X_\text{cli}(M,p,w)$ and $X_\text{ind}(M,p,a)$ exactly whenever $M$ consists of one row with entries in $\{-1,0,+1\}$ and moreover consider several cases in which $M$ consists of two rows or has arbitrary entries from $\mathbb{Z}$.