The concept of rainbow connection number of a graph was introduced by Chartrand et al. in 2008. Inspired by this concept, other concepts on colored version of connectivity in graphs were introduced, such as the monochromatic connection number by Caro and Yuster in 2011, the proper connection number by Borozan et al. in 2012, and the conflict-free connection number by Czap et al. in 2018, as well as some other variants of connection numbers later on. Chakraborty et al. proved that to compute the rainbow connection number of a graph is NP-hard. For a long time, it has been tried to fix the computational complexity for the monochromatic connection number, the proper connection number and the conflict-free connection number of a graph. However, it has not been solved yet. Only the complexity results for the strong version, i.e., the strong proper connection number and the strong conflict-free connection number, of these connection numbers were determined to be NP-hard. In this paper, we prove that to compute each of the monochromatic connection number, the proper connection number and the conflict free connection number for a graph is NP-hard. This solves a long standing problem in this field, asked in many talks of workshops and papers.

中文翻译：

---

图的三种颜色连接的硬度结果

图的彩虹连接数的概念是由 Chartrand 等人引入的。2008 年，受此概念的启发，引入了其他有关图形中连接的彩色版本的概念，例如 Caro 和 Yuster 在 2011 年提出的单色连接数，Borozan 等人提出的正确连接数。2012 年，以及 Czap 等人的无冲突连接数。在 2018 年，以及稍后的其他一些连接号变体。Chakraborty 等。证明计算图的彩虹连接数是 NP 难的。长期以来，人们一直试图解决图的单色连接数、适当连接数和无冲突连接数的计算复杂度。然而，它还没有得到解决。只有强版本的复杂性结果，即，这些连接数中的强正确连接数和强无冲突连接数被确定为NP-hard。在本文中，我们证明计算图的单色连接数、正确连接数和无冲突连接数中的每一个都是NP-hard的。这解决了该领域长期存在的问题，在研讨会和论文的许多演讲中都提出过这个问题。