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.

翻译：我们研究哈密顿环问题的以下推广：给定整数$a,b$和图$G$，是否存在$G$中的一条闭行走，使得每个顶点至少被访问$a$次且至多被访问$b$次？等价地，是否存在$2b \cdot G$的一个连通$[2a,2b]$因子，且其所有度数为偶数？对于任意常数$1 \leq a \leq b$，该问题均是NP难的。然而，已知归约所构造的图的最大度会随$b$线性增长。当$a = b = 1$时——即经典的哈密顿性问题——即使在$3$-正则图中也保持NP难；一个自然的问题是对于其他$a$, $b$是否同样成立。本文研究了在度受约束的图中，哪些$a, b$对允许多项式时间算法，哪些会导致NP难性。我们针对正则图和有向及无向的有界最大度图，给出了完整的分类刻画。