Questions tagged [hamiltonian-path]

A path in a graph that visits each vertex exactly once.

396 questions
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How many knight's tours are there?

The knight's tour is a sequence of 64 squares on a chess board, where each square is visted once, and each subsequent square can be reached from the previous by a knight's move. Tours can be cyclic, ...
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Covering pairs with permutations

Consider an $n \times n$ matrix $M_n$ with the following properties: Each row is a permutation of $A_n \equiv \{1, 2, ..., n\}$. Every ordered pair $(i,j)$, $i,j \in A_n$, $i \neq j$, appears as a ...
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Who has a winning strategy in the hamilton-circle-game?

The game starts with a graph with $n$ vertices and no edges. The players alternately add edges until the graph contains a hamilton-circle. The player who made the last move loses. Who has a winning ...
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Prove that every tournament contains at least one Hamiltonian path.

A tournament is a directed graph with exactly one edge between every pair of vertices. (So for each pair (u,v) of vertices, either the edge from u to v or from v to u exists, but not both.) ...
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Can a bipartite graph have many Hamiltonian paths but no Hamiltonian cycle?

Can a bipartite graph with at least three vertices have the following properties simultaneously: Every vertex is the initial vertex of some Hamiltonian path. The graph contains no Hamiltonian cycle. ...
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Is this graph Hamiltonian?

I know that a Hamiltonian graph has a path that visits each vertex once. But I am not sure how to figure out if this one does. Obviously I can try and trace various different paths to see if one works ...
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What is the maximum number of edges in an $n$-vertex non-Hamiltonian graph of minimum degree at least $2$?

Q: What is the maximum number of edges in an $n$-vertex non-Hamiltonian (simple) graph of minimum degree at least $2$? This question relates to Maximum number of edges in a non-Hamiltonian graph ...
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How many hamilton paths can a non-hamiltonian graph have?

What is the maximum number of hamilton paths a graph with $n$ vertices can have without having a hamiton cycle ? If my turbo pascal program works well, the first few values for $3,4,...$ vertices ...
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Does every connected $r$-regular bipartite graph contain a Hamiltonian cycle?

Here's a quickie: Let $r\ge2$. Does every connected $r$-regular bipartite graph contain a Hamiltonian cycle? I've been playing around with this for almost an hour, but I can't prove it.
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Chance of Hamiltonian Path in Sudoku cell

Checking the correctness of a daily Sudoku I'd just finished, I noticed a curious pattern in one of the 3x3 cells: 1 9 3 8 2 4 7 6 5 Note that each of the ...
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Hamiltonian Weighted Graph and Decision Problems

I ran into a question on previous Mid-Exam. anyone could clarify me? Problem A: Given a Complete Weighted Graph G, find a Hamiltonian Tour with minimum weight. Problem B: Given a Complete Weighted ...
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All 4-connected planar graphs are Hamiltonian-connected

I started reading Thomassen's paper A Theorem on Paths in Planar Graphs, where he proves one of Plummer's conjectures: Every $4$-connected planar graph is Hamiltonian-connected. Context. Recall that ...
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Is there a name for graphs with the following property?

The property of the graph is the following: For any vertex, there is a hamiltonian path starting with this vertex, but the graph is not hamiltonian. The following graph is a small example: Important ...
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Finding a Hamiltonian cycle for the set of RGB colors?

I'm consulting on an art project where a question about colors came up. Consider the RGB colors as a graph. The nodes are ordered triples $(r, g, b)$ where each entry is an integer from $[0, 255]$....
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Proving that a graph of a certain size is Hamiltonian

For any graph with order $n \geq 3$, given that its size is $$m \geq \frac{\left(n-1\right)(n-2)}{2} + 2,$$ show that the graph is Hamiltonian. I know that if I can show that the degree sum of any ...
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How many routes are there from $A$ to $B$ that cross every node exactly once?

Imagine an $n \times n$ grid, we start on one corner of the grid in square $A$, and need to reach the opposite corner to square $B$. The rules are, you can only move to an adjacent square, you can't ...
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Proof that there is no closed knight tour on a $3\ \times\ 8$ - board

I want to prove that there is no closed knight tour on a $3\ \times\ 8$ - board by deleting $s$ vertices of the corresponding knight graph to get a graph with more than $s$ connected components (...
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Is this the smallest graph with the desired properties?

The above graph has the following properties : $1$) Every vertex is start vertex of some hamiltonian path. $2$) It contains no hamiltonian cycle. $3$) It has no cycle of length $3$. $4$) It is planar....
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Homework - Proof: Is this particular graph Hamiltonian?

I have a homework for my class to Combinatorics and Graphs which I'm not sure how to finish. The task: Let G be a simple graph on 14 vertices, with 4 vertices having degree 5 and 10 vertices having ...
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The number of Hamiltonian paths in a tournament is always odd?

A tournament is defined as an orientation of a complete graph. Prove or disprove the following statement: In a tournament, there are exactly an odd number of Hamiltonian paths. In all cases I’ve ...