发表机构
AGH University of Krakow（克拉科夫AGH科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究图的在线多数边染色问题，证明在最小度满足特定条件时，贪心策略使用至多五色且为在线最优，并推广至1/k-多数边染色。
AI 中文摘要
图 $G$ 的多数边染色是对 $G$ 的边进行染色，使得对于 $G$ 的每个顶点 $v$，与 $v$ 关联的边中至多一半染有相同颜色。这一概念由 Bock 等人于 2023 年提出，他们证明了每个最小度至少为 2 的图都有多数 4-边染色。我们研究多数边染色的在线变体，其中图由演示者逐边揭示，算法必须立即且不可撤销地对每条边进行染色。特别地，我们证明了当 $\delta = O(\frac{\log n}{\log\log n})$ 时，使用至多五种颜色的贪心策略在在线算法中是最优的。我们进一步将结果推广到 $1/k$-多数边染色。
英文摘要
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 $δ= O(\frac{\log n}{\log\log n})$. We further extend our results to $1/k$-majority edge-colourings.
Comments15 pages
