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.

Chartrand等人介绍了图的彩虹连接数的概念。受此概念的启发，图形中的连通性的彩色版本也引入了其他概念，例如Caro和Yuster于2011年提出的单色连接号，Borozan等人提出了正确的连接号。在2012年，Czap等人的无冲突连接号。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，一直试图固定图形的单色连接数，适当的连接数和无冲突连接数的计算复杂度。但是，尚未解决。强版本仅会导致复杂性，即 确定这些连接号中的强正确连接号和强无冲突连接号是NP-hard。在本文中，我们证明为计算图的每个单色连接数，图的正确连接数和无冲突连接数是NP-hard。在许多研讨会和论文的讨论中，这解决了该领域长期存在的问题。