On interval colourings of graphs

TL;DR

This paper investigates the properties of interval colourings in graphs, establishing bounds on the interval thickness and confirming a conjecture about the maximum colours in planar graph colourings.

Contribution

It provides new bounds on the interval thickness of graphs and confirms a conjecture regarding the maximum number of colours in planar graph interval colourings.

Findings

Bounds on interval thickness:  rac{ ext{log} n}{ ext{log} ext{log} n} \u2264  heta(n)  n^{8/9+o(1)}

Answer to a question by Asratian, Casselgren, and Petrosyan

Confirmation of Axenovich's conjecture on planar graph colourings

Abstract

An interval colouring of a graph $G=(V,E)$ is a proper colouring $c\colon E\to \mathbb{Z}$ such that the set of colours of edges incident to any given vertex forms an interval of $\mathbb{Z}$. The interval thickness $\theta(G)$ of a graph $G$ is the smallest integer $k$ such that $G$ can be edge-partitioned into $k$ interval colourable graphs, and $\theta(n)$ is the largest interval thickness over graphs on $n$ vertices. We show that $c \frac{\log n}{\log \log n} \leq \theta(n) \leq n^{8/9+o(1)}$ for some $c>0$. In particular this answers a question by Asratian, Casselgren, and Petrosyan. In the second part of the paper, we confirm a conjecture of Axenovich that the maximum number of colours used in an interval colouring of a planar graph on $n$ vertices is at most $3n/2-2$.