On interval colourings of graphs
Lawrence Hollom, Julien Portier, Leo Versteegen

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 is a proper colouring such that the set of colours of edges incident to any given vertex forms an interval of . The interval thickness of a graph is the smallest integer such that can be edge-partitioned into interval colourable graphs, and is the largest interval thickness over graphs on vertices. We show that for some . 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 vertices is at most .
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research
