What is a Hamiltonian path?

It's a path through a graph that visits every vertex exactly once, using only edges that exist in the graph — it doesn't need to return to its starting point (that would make it a Hamiltonian cycle instead).

Why is finding a Hamiltonian path hard?

The Hamiltonian path problem is NP-complete, meaning no algorithm is known that solves it dramatically faster than trying combinations of orderings, and the number of possible orderings grows factorially with the number of nodes.

Why is this calculator limited to 10 nodes?

Backtracking search explores a search space that can grow factorially in the worst case. Ten nodes keeps that search comfortably fast for a browser-based tool, while still covering most hand-entered graph examples people want to check.

How does the backtracking search work?

It builds a path one node at a time, always extending to an unvisited neighbor; if it reaches a dead end where no unvisited neighbor is available, it backtracks (undoes the last step) and tries a different neighbor, repeating from every possible starting node.

What's the difference between a Hamiltonian path and a Hamiltonian cycle?

A Hamiltonian path just needs to visit every vertex once; a Hamiltonian cycle additionally requires the path to return to its starting vertex, forming a closed loop — both are NP-complete to detect.

Does every graph have a Hamiltonian path?

No — many graphs, especially sparse or disconnected ones, have none at all. A disconnected graph can never have a Hamiltonian path, since the path would need to jump between components with no edge to use.

How is this different from the Minimum Spanning Tree Calculator?

A minimum spanning tree connects every node using a tree structure (which can branch), while a Hamiltonian path is a single unbranching line that visits every node exactly once — a much more restrictive requirement.

Where is the Hamiltonian path problem used in real life?

It's closely related to the traveling salesman problem and appears in route planning, DNA sequencing (fragment assembly), and puzzle problems like the knight's tour on a chessboard.

Hamiltonian Path Checker — Find a Path That Visits Every Node Exactly Once

Our Hamiltonian Path Checker searches for a path through a graph that visits every node exactly once, trying each possible starting node and backtracking through possible routes, given your graph as a simple text edge list (e.g. A-B, B-C, C-D).

Quick Answer

Enter your graph as an edge list (up to 10 nodes) to instantly search for a Hamiltonian path — a route visiting every node exactly once — via backtracking.

Edges (comma-separated)

Enter edges as NodeA-NodeB, e.g. A-B, B-C, C-D. Maximum 10 nodes.

How to Use the Hamiltonian Path Checker — Visit Every Node Once Online

1
Enter your graph as a comma-separated edge list, e.g. A-B, B-C, C-D.
2
Click 'Calculate' to search for a Hamiltonian path.
3
Review the path found, or confirmation that none exists.

Why Use Hamiltonian Path Checker — Visit Every Node Once Online?

A Hamiltonian path visits every vertex in a graph exactly once — unlike an Eulerian path, which is about visiting every edge, this is about visiting every node. Determining whether one exists is NP-complete in general, meaning there's no known algorithm dramatically faster than trying out possibilities systematically, so this calculator uses backtracking search: build a path one node at a time, and backtrack whenever a dead end is reached, trying every possible starting node. Because the search space grows factorially with graph size, this calculator is deliberately scoped to at most 10 nodes, where exhaustive backtracking still completes instantly.

Frequently Asked Questions

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