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 $1\le a\le b$. 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.