Inspired by Sheehan's conjecture that no $4 $-regular graph contains exactly one
hamiltonian cycle, we prove results on hamiltonian cycles in regular graphs and nearly …

Motivated by work of Haythorpe, Thomassen and the author showed that there exists a
positive constant c such that there is an infinite family of 4-regular 4-connected graphs, each …

We study how few longest cycles a regular graph can have under additional constraints. For
each integer r≥ 5, we give exponential improvements for the best asymptotic upper bounds …

In this paper we consider the following problem: Given a Hamiltonian graph G, and a
Hamiltonian cycle C of G, can we compute a second Hamiltonian cycle C′≠ C of G, and if …

In 1999, Jacobson and Lehel conjectured that, for, every k-regular Hamiltonian graph has
cycles of many different lengths. This was further strengthened by Verstraëte, who asked …

We study how few pairwise distinct longest cycles a regular graph can have under additional
constraints. For each integer $ r\geq 5$, we give exponential improvements for the best …

arxiv:2409.17205v1 [math.CO] 25 Sep 2024 Page 1 arxiv:2409.17205v1 [math.CO] 25 Sep
2024 Cubic graphs of given girth and connectivity with a unique longest cycle Jorik Jooken1 …

In this paper we consider the following total functional problem: Given a cubic Hamiltonian
graph $ G $ and a Hamiltonian cycle $ C_0 $ of $ G $, how can we compute a second …

The protection of wavelength division multiplexed (WDM) networks can be done very
efficiently by the pre-configured protection cycles (p-cycles). The individual wavelengths …

As the Hamiltonian cycle covers all the different vertices without any repetition in a network,
it enables the decision-maker to take the decision into account for a planning. Many of the …