Hamiltonian Path Calculator

🌌 Unraveling the Mysteries of the Hamiltonian Path

Welcome to the ultimate guide on the Hamiltonian path. This fascinating concept from graph theory, named after the Irish mathematician Sir William Rowan Hamilton, is a cornerstone of computer science and discrete mathematics. It poses a simple question with a profoundly complex answer: can you visit every city on a map exactly once?

❓ What is a Hamiltonian Path? A Core Definition

A Hamiltonian path definition is straightforward: In a given graph (a set of vertices connected by edges), a Hamiltonian path is a path that visits each vertex exactly once. Think of it as a perfect tour through a network where you never revisit a location.

• Graph: A collection of points (vertices or nodes) and lines (edges) connecting them.
• Path: A sequence of vertices where each adjacent pair in the sequence is connected by an edge.
• Hamiltonian Path: A path that includes every single vertex of the graph without repetition.

If a Hamiltonian path starts and ends at the same vertex, forming a closed loop, it is called a Hamiltonian cycle or Hamiltonian circuit.

🔄 Hamiltonian Path vs Cycle: A Tale of Two Tours

The distinction between a path and a cycle is crucial. While both are fundamental to the Hamiltonian path and cycle problem, they have different properties and applications.

A classic Hamiltonian path example is the path graph Pn itself. A classic cycle example is the complete graph Kn for n ≥ 3.

🤯 The Hamiltonian Path Problem: Why It's So Hard (NP-Complete)

The Hamiltonian path problem asks whether a Hamiltonian path exists in a given graph. This problem is famously NP-complete, placing it in a class of problems for which no known efficient (polynomial-time) algorithm exists. Here’s what that means:

• NP (Nondeterministic Polynomial time): If you are given a potential solution (a path), you can quickly verify if it's a valid Hamiltonian path in polynomial time.
• NP-Complete: It's one of the "hardest" problems in NP. If you could solve the Hamiltonian path problem efficiently, you could solve all other NP problems efficiently (like the Traveling Salesman Problem, 3-SAT, etc.).

Because the Hamiltonian path is NP-complete, finding a solution for large graphs involves a brute-force search of all possible paths, which grows factorially with the number of vertices. This is why our calculator uses a backtracking Hamiltonian path algorithm, which is exhaustive but can be slow for larger graphs.

🚶‍♂️ Hamiltonian Path vs Euler Path: Vertices vs Edges

Another common point of confusion is the difference between a Hamiltonian path and an Eulerian path. The distinction is simple but fundamental:

• Hamiltonian Path: Visits every vertex exactly once.
• Eulerian Path: Traverses every edge exactly once.

A graph can have one, both, or neither. For example, the Petersen graph has a Hamiltonian path but not an Eulerian path (as it has vertices of odd degree). The existence of an Eulerian path can be determined easily by checking the degrees of the vertices, whereas the Hamiltonian path problem remains difficult.

🌐 Special Cases and Fascinating Theorems

The study of Hamiltonian paths is rich with specific cases and theorems that provide shortcuts or definitive answers for certain types of graphs.

Does the Petersen graph have a Hamiltonian path?

Yes, the famous Petersen graph does have a Hamiltonian path. However, it famously does not have a Hamiltonian cycle. This makes it a common counterexample in graph theory textbooks.

Maximum Number of Diagonal Moves in a Hamiltonian Path on a Grid

This is a fascinating combinatorial problem. When considering an n x n grid with diagonal moves allowed, finding the maximum number of diagonal moves in a Hamiltonian path is a non-trivial puzzle. The answer depends on the parity of n. For a general grid, this requires specialized algorithms to explore path configurations that prioritize diagonal movements.

Every tournament has a Hamiltonian path

This is a beautiful result known as Rédei's theorem. A tournament is a directed graph where every pair of vertices has exactly one directed edge between them (like in a round-robin tournament). Landau's theorem builds on this, stating that such a path exists, and it can be found efficiently.

Directed Hamiltonian Path

In a directed graph, a directed Hamiltonian path must follow the direction of the edges. This adds another layer of complexity but is essential for modeling processes with a specific flow.

Graphs Without Hamiltonian Paths

Not all graphs are cooperative. There are many examples of graphs that lack a Hamiltonian path:

• Polyominoes without Hamiltonian path: Certain shapes made of squares, or polyominoes, cannot be traversed with a Hamiltonian path on their corresponding grid graph.
• Cubic graph without Hamiltonian path: A cubic graph is one where every vertex has a degree of 3. The aforementioned Petersen graph is one, but the smallest is the Tutte graph with 46 vertices.
• Connected locally connected graph without Hamiltonian path: These are more esoteric examples showing that even "nice" looking graphs can fail to have a Hamiltonian path. A graph is locally connected if the neighborhood of every vertex is a connected subgraph.

⚙️ The Hamiltonian Path Algorithm in a Nutshell

The most common algorithm to solve the Hamiltonian path problem for a general graph is backtracking. Here’s how it works:

1. Start: Choose a starting vertex. Add it to the current path.
2. Explore: From the current vertex, move to an unvisited adjacent vertex. Add it to the path.
3. Check: If the path now contains all vertices, a solution is found! 🎉
4. Backtrack: If you reach a dead end (no unvisited neighbors) and the path is not yet complete, remove the current vertex from the path and go back to the previous vertex. Try a different unvisited neighbor from there.
5. Repeat: Continue this process until a path is found or all possibilities have been exhausted.

This recursive process guarantees finding a path if one exists but explores a potentially massive search space, highlighting the NP-complete nature of the problem.