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The concept of rainbow connection number of a graph was introduced byChartrand et al. in 2008. Inspired by this concept, other concepts on coloredversion of connectivity in graphs were introduced, such as the monochromaticconnection number by Caro and Yuster in 2011, the proper connection number byBorozan 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 agraph is NP-hard. For a long time, it has been tried to fix the computationalcomplexity for the monochromatic connection number, the proper connectionnumber and the conflict-free connection number of a graph. However, it has notbeen solved yet. Only the complexity results for the strong version, i.e., thestrong proper connection number and the strong conflict-free connection number,of these connection numbers were determined to be NP-hard. In this paper, weprove that to compute each of the monochromatic connection number, the properconnection number and the conflict free connection number for a graph isNP-hard. This solves a long standing problem in this field, asked in many talksof workshops and papers.

Hardness results for three kinds of colored connections of graphs