ISOMORPHIC PATH DECOMPOSITIONS OF $λK_{n,n,n} (λK^{∗}_{n,n,n})$ FOR ODD $n$

Abstract

In this paper, the isomorphic path decompositions of $\lambda$-fold balanced complete tripartite graphs $\lambda K_{n,n,n}$ and $\lambda$-fold balanced complete tripartite digraphs $\lambda K^*_{n,n,n}$ are investigated for odd $n$. We prove that the obvious necessary conditions for such decompositions in the undirected case are also sufficient; we also provide sufficient conditions for the directed case.