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# Journal of Combinatorics

## Volume 10 (2019)

### Number 1

### Revisiting the Hamiltonian theme in the square of a block: the general case

Pages: 163 – 201

DOI: http://dx.doi.org/10.4310/JOC.2019.v10.n1.a7

#### Authors

#### Abstract

This is the second part of joint research in which we show that every $2$-connected graph $G$ has the $\mathcal{F}_4$ property. That is, given distinct $x_i \in V (G), 1 \leq i \leq 4$, there is an $x_1 x_2$-hamiltonian path in $G^2$ containing different edges $x_3 y_3, x_4 y_4 \in E(G)$ for some $y_3, y_4 \in V(G)$. However, it was shown already in [3], Theorem 2, that $2$-connected DT-graphs have the $\mathcal{F}_4$ property; based on this result we generalize it to arbitrary $2$-connected graphs.We also show that these results are best possible.

#### Keywords

Hamiltonian cycles and paths, square of a block

#### 2010 Mathematics Subject Classification

05C38, 05C45

Herbert Fleischner was supported in part by FWF-grant P27615-N25.

Gek L. Chia was supported by the FRGS Grant (FP036-2013B).

Received 30 September 2015

Published 7 December 2018