Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [hamiltonicity]

For questions related to the Hamiltonicity of a graph.

A graph is Hamiltonian if it has a Hamiltonian cycle, i.e, a cycle that visits each vertex exactly once.

96 questions
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Showing that no Hamilton Circuit exists

I was confused about a certain concept and I was wondering if I could get some help. There were three points that were made in my textbook to show that a graph does not contain a Hamilton circuit: A graph with a vertex of degree one cannot have a…
Samuel Parsons
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What is the toughness of the Tutte graph?

The Tutte graph is a $3$-regular graph with $46$ vertices and $69$ edges named after W. T. Tutte. The toughness $\tau$ of a graph $G$ is $$\tau(G)=\min_{S} \left\{\frac{|S|}{c(G-S)}\right\},$$ where $c(G)$ denotes the number of connected components…
licheng
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Are there any efficient ways to tell if a graph has a Hamiltonian circuit?

For example, consider this graph. What are some common methods for determining whether the graph has a Hamiltonian circuit? After trying to find one, I'd conclude that it doesn't, but I don't know how to argue why it doesn't. Any insight on the…
user1038665
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Every cubic 3-connected Hamiltonain graph has three Hamiltonian cycles with special property

It is known that every cubic Hamiltonian graph has at least three Hamiltonian cycles (by Tutte's theorem that every edge of a cubic graph belongs to an even number of Hamiltonian cycles) Is it true that every cubic 3-connected planar Hamiltonian…
Cyriac Antony
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Construct some special non-Hamiltonian graphs.

The following theorem is well known. Theorem 1. If $G$ is a graph containing a set $S \subset V(G)$ such that $G-S$ has more than $|S|$ components, then $G$ is not Hamiltonian. We know the converse of Theorem 1 is false. We can find some examples.…
licheng
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2-connected graph is Hamiltonian

I tried to prove this: If $G$ is a $2$-connected graph with independence number $2$, then $G$ is Hamiltonian. I was thinking to construct de hamiltonian cycle. We know that $G$ has independence number $2$ then exists two vertices that are not…
TresTresUno
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Difficulty in understanding the proof of Petersen Graph is non hamiltonian as given in graph theory text by Chartrand and Zhang

I was going through the text : A First Course in Graph Theory by Chartrand and Zhang where I could not understand a few statements in the proof. Below is the excerpt: Theorem 6.4 : Petersen graph is non-Hamiltonian Proof : Suppose that the Petersen…
Abhishek Ghosh
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Hamiltonian graph on a $8\times 8$ chessboard with upper left corner and bottom right corner square removed

Suppose we are given the setup in the title. Two squares are adjacent if and only if they share a common edge. I want to find out whether the obtained graph considering squares as nodes would be Hamiltonian or not? I am out of ideas as to how to…
Sj2704
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Is There any Untraceable Generalized Petersen Graph?

The Petersen graph is one of the example of graph which is not Hamiltonian. Can we find an example among the generalized Petersen graph which doesn't have Hamiltonian path (untraceable)?
user966
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1-tough non-Hamiltonian graphs

The Petersen graph is a famous example of a 1-tough non Hamiltonian graph, and I stumbled across the following graph which also follows the property: . I found this example in a paper by V. Chvátal. However, I found out that we can extend this graph…
Kian Shah
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Consider vertex $v$ in graph $G$. Let $v$ have at least three adjacent vertices with degrees of two. Prove that $G$ is not a Hamiltonian graph.

Consider vertex $v$ in graph $G$. Let $v$ have at least three adjacent vertices with degrees of two. Prove that $G$ is not a Hamiltonian graph. Proof Suppose $G$ is Hamiltonian. Let $x$, $y$ and $z$ be the vertex adjacent to $v$. We know that $G$…
t-y
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Proving a graph does not have a hamilton path

I can see intuitively why this graph will not have a Hamilton path, but I can't seem to write up a convincing proof. Are there any tips on how to prove that a graph does not have a hamilton path?
Ray
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How many Distinct Hamiltonian Maximal Planar Graphs are there (n vertices) and could this representation help?

If we make a regular polygon with n vertices (n edges) and triangulate on the inside with n-3 edges, then triangulate on the outside with (n-3) edges (or draw dotted lines inside again), a Maximal Planar Graph is formed. Edges shouldn't be repeated…
John Hunter
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Using Bellman-Ford to find a Hamiltonian cycle? (NP-complete)

Let $G(V,E)$ be a directed graph, where $V=\{a_1,\ldots,a_n\}$ is a set of vertices and $E$ is a set of ordered pairs of $V$, with $|V|=n$. Now, let be $G(W,F)$ be a graph where $W$ is a set of vertices, such that $W=\{a_1,\ldots,a_{2n}\}$ with…
yugikaiba
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"Tricky" questions on graph theory

So , I am revising graph theory. I need to gain some help/ feedback for those, because at least to me they are tricky. They are supposed to be answered quickly , because they come from a tight timed - exam.( so I guess they come with either some…
tonythestark
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