  In this paper, we investigate the well-studied Hamiltonian cycle problem
(HCYCLE), and present an interesting dichotomy result on split graphs. T.
Akiyama et al. (1980) have shown that HCYCLE is NP-complete in planar bipartite
graphs with maximum degree $3$. Using this reduction, we show that HCYCLE is
NP-complete in split graphs. In particular, we show that the problem is
NP-complete in $K_{1,5}$-free split graphs. Further, we present polynomial-time
algorithms for Hamiltonian cycle in $K_{1,3}$-free and $K_{1,4}$-free split
graphs. We believe that the structural results presented in this paper can be
used to show similar dichotomy result for Hamiltonian path problem (HPATH) and
other variants of HCYCLE.
