Decomposition of Complete Tripartite Graphs into Gregarious 3-Paths and 6-Cycles

In this paper, we refer to a decomposition of a tripartite graph into paths of length \( 3 \), or into \( 6 \)-cycles, as gregarious if each subgraph has at least one vertex in each of the three partite sets. For a tripartite graph to have a \( 6 \)-cycle decomposition it is straightforward to see that all three parts must have even size. Here we provide a gregarious decomposition of a complete tripartite graph \( K(r, s, t) \) into paths of length \( 3 \), and thus of \( K(2r, 2s, 2t) \) into gregarious \( 6 \)-cycles, in all possible cases, when the straightforward necessary conditions on \( r, s, t \) are satisfied.