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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What is the densest bipartite graph with unique Hamiltonian cycle?
In a prior post regarding perfect matching, it was stated that the densest graph with a unique perfect matching cannot have more than $n^2$ edges, if graph has $2n$ vertices.
Analogously, what is the ...
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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 ...
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Simple decomposition of $K_{2n}-I$ into hamiltonian cycles
http://mathworld.wolfram.com/HamiltonDecomposition.html
In the 1890s, Walecki showed that complete graphs K_n admit a Hamilton decomposition for odd n, and decompositions into Hamiltonian cycles ...
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Refinement of Dirac's theorem on Hamiltonian graphs
Dirac's theorem states that if degree of each vertex of a graph $G=(V,E)$ is not less than $|V|/2$, then it has Hamiltonian cycle. It is less known, but still known and not so hard to prove (though I ...
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Can we find 3 disjoint directed Hamiltonian cycles in the cube?
Let $D$ be the digraph on $2^d$ vertices with $d2^d$ edges that we obtain by directing each edge of the $d$-dimensional hypercube in both directions.
Can we partition the edges of $D$ into $d$ ...
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A digraph related to permutations
A finite sequence of distinct real numbers of length $n$ determines a linear order of $\{1,\ldots,n\}$, by mapping position to rank; call this the permutation of the sequence.
Consider the following ...
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Hamiltonicity of random graphs with high girth
We say that $G\sim G_{n,f}$ (for $f=f(n)$) if $G$ is chosen uniformly at random from all graphs on $n$ vertices with girth $g(G)\ge f(n)$. Is there any threshold function $F(n)$ such that when $f\ll F$...
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How to efficiently find a Hamiltonian cycle in a graph whose closure is complete?
A graph whose closure is the complete graph is Hamiltonian by the Bondy-Chvátal theorem, but I haven't found a polynomial algorithm for finding a Hamiltonian cycle in such a graph. Is there one that ...
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Hamiltonicity criteria for sparse graphs
Given a sparse graph, how can one go about proving that it is Hamiltonian? (Assuming it actually is, of course).
There are three main classes of criteria for Hamiltonicity that I am aware of:
Dirac-...
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Regular graphs with $a$ and $b$ Hamiltonian edges
Special case of this question.
Let $G$ be $r$-regular Hamiltonian graph.
An $a$ edge is an edge which is on every Hamiltonian cycle.
A $b$ edge is an edge which is on no Hamiltonian cycle.
$a(G)$ ...
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Minimum distance between Hamiltonian cycles in cubic Hamiltonian graph
It is $NP$-hard to find constant factor approximation of longest cycle in cubic Hamiltonian graphs. Therefore, finding a Hamiltonian cycle in a cubic Hamiltonian graph is NP-hard.
By Smith's theorem, ...
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Reconstructing the number of Hamiltonian cycles
As is common terminology in graph reconstruction, given a graph $G$, we call a vertex deleted subgraph of $G$, a card, and call the multiset of all cards, the deck of $G$. The graph reconstruction ...
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Details about Kelman's equivalent form of Barnette's conjecture
Barnette's conjecture states that every cubic planar bipartite 3-connected graph admits Hamiltonian cycles.
Kelman claims that this conjecture is equivalent to a stronger one, which imposes some ...
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Effect of removing a Hamiltonian cycle on the Laplacian spectrum
Notation: $\lambda_{\max}(G)$ is the largest eigenvalue of the Laplacian matrix of the graph $G$ (aka the Laplacian index of $G$).
Now suppose $G$ is a Hamiltonian graph with Hamiltonian cycle $C$.
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Hamiltonian cycles in power-graphs
I've stumbled across a short note from 1993 where a nice question was asked: Suppose you take a graph with vertices $\{1,2,\ldots,n\}$ and connect $i,j$ by an edge if and only if $i+j$ is a $k$th ...
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