The power of many colours
Noga Alon, Matija Buci\'c, Micha Christoph, Michael Krivelevich

TL;DR
This paper investigates the guaranteed size of monochromatic or multi-coloured connected components in edge-coloured complete graphs, extending classical problems and providing new bounds and disproving some conjectures.
Contribution
It completes the understanding of the size of guaranteed connected components in multi-colourings, solving open problems and refining previous bounds.
Findings
Determined the approximate size of guaranteed connected components up to a logarithmic factor.
Provided precise results for specific regimes, solving a problem by Liu, Morris, and Prince.
Disproved a conjecture posed by Liu, Morris, and Prince.
Abstract
A classical problem, due to Gerencs\'er and Gy\'arf\'as from 1967, asks how large a monochromatic connected component can we guarantee in any -edge colouring of ? We consider how big a connected component can we guarantee in any -edge colouring of if we allow ourselves to use up to colours. This is actually an instance of a more general question of Bollob\'as from about 20 years ago which asks for a -connected subgraph in the same setting. We complete the picture in terms of the approximate behaviour of the answer by determining it up to a logarithmic term, provided is large enough. We obtain more precise results for certain regimes which solve a problem of Liu, Morris and Prince from 2007, as well as disprove a conjecture they pose in a strong form. We also consider a generalisation in a similar direction of a question first considered by Erd\H{o}s and…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsLimits and Structures in Graph Theory
