On the spanning structure hierarchy of 3-connected planar graphs

TL;DR

This paper investigates the relationship between spanning good even cacti and prism-hamiltonicity in 3-connected planar graphs, providing counterexamples to a previously posed equivalence and exploring structural properties.

Contribution

It demonstrates that not all 3-connected planar prism-hamiltonian graphs have a spanning good even cactus, answering a question by pacapan negatively, and introduces new classes of graphs with specific spanning properties.

Findings

Existence of infinitely many 3-connected planar prism-hamiltonian graphs without spanning good even cacti.

Existence of 3-connected planar graphs with spanning good even cacti but no such cactus with maximum degree three.

Abstract

The prism over a graph $G$ is the Cartesian product of $G$ with the complete graph $K_2$. $G$ is prism-hamiltonian if the prism over $G$ has a Hamilton cycle. A good even cactus is a connected graph in which every block is either an edge or an even cycle, and every vertex is contained in at most two blocks. It is known that good even cacti are prism-hamiltonian. Indeed, showing the existence of a spanning good even cactus has become one of the most common techniques in proving prism-hamiltonicity. \v{S}pacapan asked whether having a spanning good even cactus is equivalent to having a hamiltonian prism for 3-connected planar graphs. In this article we give a negative answer to this question by showing that there are infinitely many 3-connected planar prism-hamiltonian graphs that have no spanning good even cactus. We also prove the existence of an infinite class of 3-connected planar graphs that have a spanning good even cactus but no spanning good even cactus with maximum degree three.