PreprintCC BY-SA 4.0arXiv
On-line majority edge-colourings of graphs
Abstract
A majority edge-colouring of a graph G is a colouring of the edges of G such that, for every vertex v of G, at most half of the edges incident with v receive the same colour. This notion was introduced by Bock et al. in 2023, who proved that every graph of minimum degree at least $2$ has a majority $4$-edge-colouring. We investigate an on-line variant of majority edge-colouring in which the graph is revealed by the Presenter edge-by-edge and the Algorithm must colour each edge immediately and irrevocably. In particular. we prove that the greedy strategy, which uses at most five colours, is optimal among on-line algorithms if δ= O((log n)/(loglog n)). We further extend our results to 1/k-majority edge-colourings.
Related
BibTeX
@misc{pekala2026online,
title = {{On-line majority edge-colourings of graphs}},
author = {Paweł Pękała},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37973},
eprint = {2609.37973},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37973},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - On-line majority edge-colourings of graphs
AU - Paweł Pękała
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37973
UR - https://arxiv.org/abs/2609.37973
AB - A majority edge-colouring of a graph G is a colouring of the edges of G such that, for every vertex v of G, at most half of the edges incident with v receive the same colour. This notion was introduced by Bock et al. in 2023, who proved that every graph of minimum degree at least $2$ has a majority $4$-edge-colouring. We investigate an on-line variant of majority edge-colouring in which the graph is revealed by the Presenter edge-by-edge and the Algorithm must colour each edge immediately and irrevocably. In particular. we prove that the greedy strategy, which uses at most five colours, is optimal among on-line algorithms if δ= O((log n)/(loglog n)). We further extend our results to 1/k-majority edge-colourings.
ER -