BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:Modular Graph Colourings - Gaia Carenini (Cambridge)
DTSTART:20251030T143000Z
DTEND:20251030T153000Z
UID:TALK240079@talks.cam.ac.uk
CONTACT:103978
DESCRIPTION:Given a graph G and an integer k ≥ 2\, let χ′ₖ(G) denot
 e the minimum number of colours required to colour the edges of G so that\
 , in each colour class\, the subgraph induced by the edges of that colour 
 has all non-zero degrees congruent to 1 modulo k. In 1992\, Pyber proved t
 hat χ′₂(G) ≤ 4 for every graph G and asked whether χ′ₖ(G) can 
 be bounded solely in terms of k for every k ≥ 3. This question was answe
 red in 1997 by Scott\, who showed that χ′ₖ(G) ≤ 5k² log k\, and fu
 rther asked whether χ′ₖ(G) grows only linearly with k. Recently\, Bot
 ler\, Colucci\, and Kohayakawa (2023) answered Scott’s question affirmat
 ively\, proving that χ′ₖ(G) ≤ 198k − 101\, and conjectured that t
 he multiplicative constant could be reduced to 1. A step toward this conje
 cture was made in 2024 by Nweit and Yang\, who improved the bound to χ′
 ₖ(G) ≤ 177k − 93.In this work\, we further improve the multiplicativ
 e constant to 9. More specifically\, we show that χ′ₖ(G) ≤ 7k + o(k
 ) when k is odd\, and χ′ₖ(G) ≤ 9k + o(k) when k is even. As part of
  our proof\, we establish that χ′ₖ(G) ≤ k + O(d) for every d-degene
 rate graph G\, a result that plays a central role in our argument.
LOCATION:MR12
END:VEVENT
END:VCALENDAR