On the Hamiltonian Property Hierarchy of 3-Connected Planar Graphs

The prism over a graph $G$ is the Cartesian product of with complete $K_2$. prism-hamiltonian if has Hamilton cycle. A good even cactus connected in which every block either an edge or cycle and vertex contained at most two blocks. It known that cacti are prism-hamiltonian. Indeed, showing existence spanning become common technique proving prism-hamiltonicity. Špacapan [S. Špacapan. counterexample to prism-hamiltonicity 3-connected planar graphs. J. Combin. Theory Ser. B, 146:364--371, 2021] asked whether having equivalent hamiltonian for In this article we answer his question negative, by there infinitely many graphs have no cactus. addition, prove infinite class but maximum degree three.