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 · G with all degrees even? This problem is NP-hard for any constants 1 ≤ a ≤ 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.