通过边排序区分相邻顶点
Distinguishing adjacent vertices by ordering edges
- 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总结：
本文研究通过边排序区分相邻顶点的问题，证明在特定条件下（如连通非正则图或第二类图）存在全局边序使序列区分相邻顶点，并对高度正则图给出概率性结论。
AI中文摘要：
1-2-3 猜想指出，对于每个没有孤立边的图，存在一个从集合 {1,2,3} 中取值的边加权，使得相邻顶点在其关联边上的权重之和不同。在序列变体中，相邻顶点要通过其关联边上的权重序列来区分。本文研究：对于每个没有孤立边的图以及一个固定的正常边着色（其中颜色被解释为权重），是否存在一个边的全局全序，使得由此产生的关联权重序列能区分相邻顶点。对于连通图，我们证明只要存在两个相邻顶点具有不同的关联权重集合，这样的序就存在。这为连通非正则图或第二类连通图的每个正常边着色给出了肯定答案。此外，一个概率论证对度至少为六的连通正则图给出了相同的结论。
英文摘要：
The 1-2-3 Conjecture states that for every graph without isolated edges, there exists an edge-weighting from $\{1,2,3\}$ such that adjacent vertices receive distinct sums of weights on their incident edges. In the sequence variant, adjacent vertices are to be distinguished by the sequences of weights on their incident edges. In this paper, we investigate whether, for every graph without isolated edges and for a fixed proper edge colouring (where colours are interpreted as weights), there exists a global total order of the edges such that the resulting sequences of incident weights distinguish adjacent vertices. For connected graphs, we prove that such an order exists whenever there exist two adjacent vertices that have distinct sets of incident weights. This yields a positive answer for every proper edge colouring of a connected non-regular graph or a connected graph of class two. Moreover, a probabilistic argument gives the same conclusion for connected regular graphs of degree at least six.
