Complexity of Multiple-Hamiltonicity in Graphs of Bounded Degree

TL;DR

This work studies the algorithmic complexity of the generalized Hamiltonian walk problem $[a,b]\hHAM$, which asks for a closed walk visiting each vertex at least $a$ and at most $b$ times. The authors develop a unified framework using depletor gadgets and edge-labelings to characterize when the problem is solvable in polynomial time versus NP-hard, across regular graphs, bounded-degree graphs, and directed graphs. They obtain tight thresholds: in $d$-regular undirected graphs, hardness occurs iff $d$ is even and $b<d/2$, or $d$ is odd and $b<d$; in max-degree $d$ graphs, hardness arises when $a=1$ and $b<d$ or when $a>1$ and $\frac{b}{a}<d-1$; in directed graphs, max-degree $3$ yields a sharp dichotomy based on $a,b$, while max-degree $4$ is hard for all $a,b$. The main methods combine depletor gadgets with a labeling view that reduces the problem to finding feasible nonzero edge-labelings subject to connectivity, enabling reductions from classic Hamiltonian problems. The results provide a comprehensive map of tractability for $[a,b]\hHAM$ under bounded-degree constraints and lay groundwork for extensions to grids and approximation questions.

Abstract

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.

Figures (14)

• Figure 1: Reducing $[1,1]\hHAM$ to $[3,3]\hHAM$.
• Figure 2: Labellings corresponding to $[b,b]$ walks on 4-regular and 5-regular depletors, respectively.
• Figure 3: Reduction from $[1,1]\hHAM$ in $3$-regular graphs to $[4,4]\hHAM$ in $5$-regular graphs. Red outgoing edges represent depletors.
• Figure 4: The two possible labelings of the edge gadget.
• Figure 5: The edge gadgets labeled $1,1,1$ must correspond to a Hamiltonian path.

Theorems & Definitions (45)

• Definition 1
• Lemma 1: Broersma, Göbel broersmakwalks
• proof
• Lemma 2
• proof
• Lemma 3: Broersma, Göbel broersmakwalks
• proof
• Lemma 4
• proof
• Lemma 5