A degree sum condition on Hamiltonian cycles in balanced 3-partite graphs (Q1377822)

Let \(G\) be a balanced tripartite graph of order \(3n\). It is shown that if for each nonadjacent pair of vertices \(u\) and \(v\) in different parts, \(d_H(u)+d_H(v)\geq n+1\), where \(H\) is the bipartite graph spanned by the two parts containing \(u\) and \(v\), then \(G\) is Hamiltonian. Similar results on sum of degree conditions in balanced \(k\)-partite graphs that imply Hamiltonicity can be found in a paper by \textit{G. Chen} and \textit{M. S. Jacobson} [Degree sum conditions for Hamiltonicity on \(k\)-partite graphs, Graphs Comb. 13, No. 4, 325-343 (1997; Zbl 0886.05087)].