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Questions tagged [hamiltonian-graphs]

A Hamiltonian graph (directed or undirected) is a graph that contains a Hamiltonian cycle, that is, a cycle that visits every vertex exactly once.

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Does this graph contain at least two Hamiltonian cycles?

Let $G$ be a simple graph which is a $2n$-cycle together with $n$ chords such that $G$ is $3$-regular. In other words, the set of $n$ chords is a perfect matching of $G$. I conjecture that for every ...
user40096's user avatar
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16 votes
1 answer
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A counterexample to a conjecture of Nash-Williams about hamiltonicity of digraphs?

Maybe I am missing something, but found potential counterexample to a conjecture of Nash-Williams. According to HAMILTONIAN DEGREE SEQUENCES IN DIGRAPHS The outdegree and indegree sequences of ...
joro's user avatar
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10 votes
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Graphs with many edges avoided by Hamiltonian cycles

Let $G$ be a $3$-connected Hamiltonian graph with at least one edge that belongs to each H-cycle of $G$. Some authors (e.g. in the link given here) call such an edge an a-edge and an edge that belongs ...
Wolfgang's user avatar
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15 votes
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What is the smallest uniquely hamiltonian graph with minimum degree at least 3?

I would like to know more about uniquely hamiltonian graphs with minimum vertex degree at least 3, and in particular what is the smallest one. (Recall that a graph is hamiltonian if it has a cycle ...
Gordon Royle's user avatar
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Generalisation of this circular arrangement of numbers from $1$ to $32$ with two adjacent numbers being perfect squares

I posted this question on MSE, and failed to get the type of answer I wanted. That's why I would like to post it here and wait for the experts to reply. Here's the link to the MSE post, which I ...
Sayan Dutta's user avatar
7 votes
7 answers
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Efficient Hamiltonian cycle algorithms for graph classes

Generally speaking, finding a Hamiltonian cycle is NP-Hard and so tough. But if $G=L(H)$ is the line graph of $H$, then we can reduce the problem of finding a Hamiltonian cycle in $G$ to finding an ...
Felix Goldberg's user avatar
5 votes
1 answer
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Infinitely many counterexamples to Nash-Williams's conjecture about hamiltonicity?

Question from 2013 gives one counterexample to Nash-Williams's conjecture about hamiltonicity of dense digraphs. Later, we found tens of counterexamples on more than 30 vertices and believe there are ...
joro's user avatar
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Graph gadget related to uniquely hamiltionian regular graphs

A graph is uniquely hamiltonian if it has exactly one hamiltonian cycle. According to a conjecture there are no $r$-regular uniquely hamiltonian graphs for $r > 2$ and of special interest is the ...
joro's user avatar
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3 votes
1 answer
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Properties of Hamilton cycles in hypercubes

Questions: is the following true? for $n\in\mathbb{N}$ every Hamilton cycle in an $n$-dimensional hypercube $Q_n$ there exist $2^{n-1}$ edges that are mutually parallel $Q_2$ is the only case in ...
Manfred Weis's user avatar
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2 votes
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Two more counterexamples to a conjecture from 1975 about hamiltonicity of digraphs

Question from 2013 gives one counterexample to Nash-Williams's conjecture 1975 about hamiltonicity of dense digraphs. In the linked answer, @LouisD "reverse engineered" the counterexample ...
joro's user avatar
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