Long cycles and long paths in the Kronecker product of a cycle and a tree Original Research Article

Let Cm × T denote the Kronecker product of a cycle Cm and a tree T. If m is odd, then Cm × T is connected, otherwise this graph consists of two isomorphic components. This paper presents a scheme which constructs a long cycle in each component of Cm × T. If T satisfies certain degree constraints, then the cycle thus traced is shown to be a dominating set, and in some cases, a vertex cover of that component. The procedure builds on (i) results on longest cycles in Cm × Pn, and (ii) a path factor of T. Additional results include characterizations for the existence of a Hamiltonian cycle and for that of a Hamiltonian path in Cm × T.