On the Burkard–Hammer condition for hamiltonian split graphs Original Research Article

A graph G=(V,E)G=(V,E) is called a split graph if there exists a partition V=I∪KV=I∪K such that the subgraphs of G induced by I and K are empty and complete graphs, respectively. In 1980, Burkard and Hammer gave a necessary but not sufficient condition for hamiltonian split graphs with |I|<|K||I|<|K|. In this paper, we show that the Burkard–Hammer condition is also sufficient for the existence of a Hamilton cycle in a split graph G such that 5≠|I|<|K|5≠|I|<|K| and the minimum degree δ(G)⩾|I|-3δ(G)⩾|I|-3. For the case 5=|I|<|K|5=|I|<|K|, all split graphs satisfying the Burkard–Hammer condition but having no Hamilton cycles are also described.