关于使用少数颜色的强多数边着色
On Strong Majority Edge Colourings with Few Colours
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中文总结 AI 辅助
该研究针对图的强多数边着色问题，证明最小度δ≥5的图可使用3种颜色完成着色，改进了此前的边界，并引入强1/k-多数边着色框架建立对应边界。
中文摘要 AI 辅助
图G的强多数边着色是一种边着色，其中对于每条边e和每种颜色α，与e相邻的边中最多有一半被赋予颜色α。与许多相关的着色概念一样，它可以自然地解释为关联超图的着色问题。令人惊讶的是，尽管对应的超图可能具有任意大的顶点度，但在适度的最小度假设下，强多数边着色所需颜色数存在一个通用的有限上界，这与其他几个密切相关的多数概念不同。我们特别证明，每个最小度δ≥5的图G都可以使用三种颜色进行强多数边着色，这改进了之前已知的三种颜色要求δ≥9的边界，以及δ≥5时四种颜色足够的结果。我们的结果在颜色数方面是最优的，在最小度假设方面也接近最优，因为我们证明最小度不能降低到4以下。我们还引入了更一般的强1/k-多数边着色框架，并为该设置建立了相应的边界。
英文摘要
A strong majority edge colouring of a graph $G$ is an edge colouring in which, for every edge $e$ and every colour $α$, at most half the edges adjacent to $e$ receive colour $α$. Like many related colouring notions, it admits a natural interpretation as a colouring problem for an associated hypergraph. Somewhat surprisingly, although the corresponding hypergraph may have arbitrarily large vertex degrees, a universal finite upper bound on the sufficient number of colours in a strong majority edge colouring exists under a natural modest minimum degree assumption, unlike in several other closely related majority concepts. We in particular prove that every graph $G$ with minimum degree $δ\ge5$ admits a strong majority edge colouring with three colours, improving both the previously known bound $δ\ge9$ for three colours and the result showing that four colours suffice whenever $δ\ge5$. Our result is best possible with respect to the number of colours and the minimum degree assumption. We also introduce a more general framework of strong $1/k$-majority edge colourings and establish corresponding bounds for this setting.
