Hardness results for three kinds of colored connections of graphs

TL;DR

This paper proves that computing the monochromatic, proper, and conflict-free connection numbers of a graph is NP-hard, resolving a long-standing open problem in graph theory related to colored connectivity measures.

Contribution

It establishes the NP-hardness of calculating three key colored connection numbers, advancing understanding of their computational complexity.

Findings

NP-hardness of monochromatic connection number

NP-hardness of proper connection number

NP-hardness of conflict-free connection number

Abstract

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.