On strongly hamiltonian abelian group graphs

Abstract

Let G be a graph containing a spanning subgraph H isomorphic to C_mxL_n where C_m denotes the cycle with m vertices (m≥3) and L_n the path with n, vertices (n≥2). Then H, in turn, contains a spanning subgraph H′ isomorphic to L_mxL_n. Vertices in H′ can thus be coloured by two colours, say blue and red, so that no two adjacent vertices in H′ are of the same colour. Then any two vertices in G are connected by a hamiltonian path if and only if G contains an edge joining two blue vertices and an edge joining two red vertices. This result enables us to characterize abelian group graphs G in whichany two vertices are connected by a hamiltonian path.