Complexity of Multiple-Hamiltonicity in Graphs of Bounded Degree
<|reference_start|>Complexity of Multiple-Hamiltonicity in Graphs of Bounded Degree: We study the following generalization of the Hamiltonian cycle problem: Given integers $a,b$ and graph $G$, does there exist a closed walk in $G$ that visits every vertex at least $a$ times and at most $b$ times? Equivalently, does there exist a connected $[2a,2b]$ factor of $2b \cdot G$ with all degrees even? This problem is NP-hard for any constants $1 \leq a \leq b$. However, the graphs produced by known reductions have maximum degree growing linearly in $b$. The case $a = b = 1 $ -- i.e. Hamiltonicity -- remains NP-hard even in $3$-regular graphs; a natural question is whether this is true for other $a$, $b$.   In this work, we study which $a, b$ permit polynomial time algorithms and which lead to NP-hardness in graphs with constrained degrees. We give tight characterizations for regular graphs and graphs of bounded max-degree, both directed and undirected.<|reference_end|>
@article{liu2024complexity,
    title={Complexity of Multiple-Hamiltonicity in Graphs of Bounded Degree},
    author={Brian Liu, Nathan S. Sheffield, Alek Westover},
    journal={arXiv preprint arXiv:2405.16270},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.16270},
    primaryClass={cs.CC}
}