A note on interval colourings of graphs

• Main Authors: Axenovich, Maria, Girão, António, Hollom, Lawrence, Portier, Julien, Powierski, Emil, Savery, Michael, Tamitegama, Youri, Versteegen, Leo
• Format: Journal Article
• Language: English
• Published: 08.03.2023
• Subjects: Mathematics - Combinatorics
• DOI: 10.48550/arxiv.2303.04782

European Journal of Combinatorics, 120 (2024) A graph is said to be interval colourable if it admits a proper edge-colouring using palette $\mathbb{N}$ in which the set of colours 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 $\theta(n)$, the maximum interval colouring thickness of an $n$-vertex graph, satisfies $\theta(n) =\Omega(\log(n)/\log\log(n))$ and $\theta(n)\leq n^{5/6+o(1)}$, which improves on the trivial lower bound and an upper bound of 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.