Long Cycles and Long Paths in the Kronecker Product of a Cycle, a Tree

Let C m T denote the Kronecker product of a cycle C m and a tree T. If m is odd, then C m 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 C m T, where T satisses certain degree constraints. The cycle thus traced is shown to be a dominating set, and in some cases, a vertex cover of that component. The algorithm builds on (i) results on longest cycles in C m P n , where P n is a path on n vertices, and (ii) a path factorization of T. Additional results include characterizations for the existence of a Hamiltonian cycle and for that of a Hamiltonian path in C m T.