We show that for every graph without isolated edge, the edges can be assigned weights from $\{1,2,3\}$ so that no two neighbors receive the same sum of incident edge weights. This solves a conjecture of Karoński, Łuczak, and Thomason from 2004.

中文翻译：

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1-2-3猜想的一个解

我们证明，对于每个没有孤立边的图，可以为边分配权重$\{1,2,3\}$以便没有两个邻居收到相同的入射边权重之和。这解决了 Karoński、Łuczak 和 Thomason 2004 年的猜想。