A note on interval colourings of graphs

Abstract

A graph is said to be interval colourable if it admits a proper edge-colouring using palette $N$ in which the set of colours of edges that are incident to each vertex is an interval. The interval colouring thickness of a graph $G$ is the minimum $k$ such that $G$ can be edge-decomposed into $k$ interval colourable graphs. We show that $\theta \left(n\right)$, the maximum interval colouring thickness of an $n$-vertex graph, satisfies $\theta \left(n\right)=\Omega (log\left(n\right)/loglog\left(n\right))$ and $\theta \left(n\right)⩽n^{5/6+o\left(1\right)}$, which improves on the trivial lower bound and the upper bound given by the first author and Zheng. As a corollary, we answer a question of Asratian, Casselgren, and Petrosyan and disprove a conjecture of Borowiecka-Olszewska, Drgas-Burchardt, Javier-Nol, and Zuazua. We also confirm a conjecture of the first author that any interval colouring of an $n$-vertex planar graph uses at most $3n/2-2$ colours.