Hardness result for the total rainbow $k$-connection of graphs

TL;DR

This paper investigates the computational complexity of the total rainbow $k$-connection number in graphs, proving that deciding whether this number equals 3 is NP-complete, highlighting the problem's computational difficulty.

Contribution

The paper establishes the NP-completeness of determining whether the total rainbow $k$-connection number equals 3 in graphs, a novel complexity result in graph coloring.

Findings

Deciding if $trc_k(G)=3$ is NP-complete.

Total rainbow $k$-connection number computation is computationally hard.

Highlights the complexity of total rainbow connectivity problems.

Abstract

A path in a total-colored graph is called \emph{total rainbow} if its edges and internal vertices have distinct colors. For an $\ell$-connected graph $G$ and an integer $k$ with $1\leq k \leq\ell$, the \emph{total rainbow $k$-connection number} of $G$, denoted by $trc_k(G)$, is the minimum number of colors used in a total coloring of $G$ to make $G$ \emph{total rainbow $k$-connected}, that is, any two vertices of $G$ are connected by $k$ internally vertex-disjoint total rainbow paths. In this paper, we study the computational complexity of total rainbow $k$-connection number of graphs. We show that it is NP-complete to decide whether $trc_k(G)=3$.