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

Abstract

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. 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 C m × P n , 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 C m × T.