Revisiting the Hamiltonian Theme in the Square of a Block: The General Case

TL;DR

This paper proves that every 2-connected graph has the ${\

Contribution

It generalizes the ${\cal F}_4$ property from DT-graphs to all 2-connected graphs, establishing the property for a broader class of graphs.

Findings

Every 2-connected graph has the ${\cal F}_4$ property.

The results are optimal and cannot be improved.

The property extends previous results on DT-graphs.

Abstract

This is the second part of joint research in which we show that every $2$-connected graph $G$ has the ${\cal F}_4$ property. That is, given distinct $x_i\in V(G)$, $1\leq i\leq 4$, there is an $x_1x_2$-hamiltonian path in $G^2$ containing different edges $x_3y_3, x_4y_4\in E(G)$ for some $y_3,y_4\in V(G)$. However, it was shown already in \cite[Theorem 2]{cf1:refer} that 2-connected DT-graphs have the ${\cal F}_4$ property; based on this result we generalize it to arbitrary $2$-connected graphs. We also show that these results are best possible.