On interval colourings of graphs

Abstract

An interval colouring of a graph G = (V, E) is a proper colouring c : E → Z such that the set of colours of edges incident to any given vertex forms an interval of Z. The interval thickness θ(G) of a graph G is the smallest integer k such that G can be edge-partitioned into k interval colourable graphs, and θ(n) is the largest interval thickness over graphs on n vertices. We show that c log n log log n ≤ θ(n) ≤ 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.