The well-known 1–2–3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with 1, 2 and 3 so that adjacent vertices receive distinct weighted degrees. This is open in general, while it is known to be possible from the weight set $\{1,2,3,4,5\}$. We show that for regular graphs it is sufficient to use weights 1, 2, 3, 4. Moreover, we prove the conjecture to hold for every d-regular graph with $d\ge {10}^8$.

中文翻译：

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规则图形的1–2–3猜想几乎成立

众所周知的1–2–3猜想断言，可以用1、2和3对每个没有孤立边的图的边进行加权，以便相邻顶点获得不同的加权度。总体上是开放的，虽然从重量设定中可以知道$\{\mathrm{1个}，2，3，4，5\}$。我们证明对于正则图，使用权重1、2、3、4足够了。此外，我们证明了对于每个d正则图都成立的猜想$d\ge {10}^8$。