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 θ(n), the maximum interval colouring thickness of an n-vertex graph, satisfies θ(n)=Ω(log(n)/loglog(n)) and θ(n)⩽n5/6+o(1), 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.