论文标题
进一步的乘法1-2-3猜想的证据
Further Evidence Towards the Multiplicative 1-2-3 Conjecture
论文作者
论文摘要
2012年由Skowronek-Kazi {ó} W介绍的1-2-3猜想的产品版本指出，除了一些显而易见的例外，所有图形都可以是3范围标记的，因此没有两个相邻的顶点会出现在同一标签的产品中。迄今为止，该猜想主要用于完整的图形和3色图。作为对猜想的强烈支持，也证明所有图都承认了这样的4个标记。在这项工作中，我们调查了如何通过Vu {\ v C} Kovi {ć}对1-2-3猜想的多版本的最新证明进行调整，以证明在产品版本上证明结果。我们证明，4个色素图验证了1-2-3猜想的产品版本。我们还证明，对于所有图表，我们都可以设计几乎具有所需属性的三个标记。这导致了一个新问题，我们为某些图形类解决了。
The product version of the 1-2-3 Conjecture, introduced by Skowronek-Kazi{ó}w in 2012, states that, a few obvious exceptions apart, all graphs can be 3-edge-labelled so that no two adjacent vertices get incident to the same product of labels. To date, this conjecture was mainly verified for complete graphs and 3-colourable graphs. As a strong support to the conjecture, it was also proved that all graphs admit such 4-labellings. In this work, we investigate how a recent proof of the multiset version of the 1-2-3 Conjecture by Vu{\v c}kovi{ć} can be adapted to prove results on the product version. We prove that 4-chromatic graphs verify the product version of the 1-2-3 Conjecture. We also prove that for all graphs we can design 3-labellings that almost have the desired property. This leads to a new problem, that we solve for some graph classes.
