The most general structure of graphs with hamiltonian or hamiltonian connected square

TL;DR

This paper characterizes the most general block-cutvertex structures of graphs whose squares are guaranteed to be hamiltonian or hamiltonian connected, extending previous results on hamiltonian total graphs.

Contribution

It determines the broadest class of graphs with specific block-cutvertex structures ensuring hamiltonicity or hamiltonian connectedness of their squares.

Findings

Identifies the general block-cutvertex structures guaranteeing hamiltonian square

Extends previous results from hamiltonian total graphs to broader classes

Provides a comprehensive characterization of graphs with hamiltonian square properties

Abstract

On the basis of recent results on hamiltonicity and hamiltonian connectedness in the square of a 2-block, we determine the most general block-cutvertex structure a graph $G$ may have in order to guarantee that $G^2$ is hamiltonian, hamiltonian connected, respectively. Such an approach was already developed for hamiltonian total graphs.