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Results tagged with hamiltonian-graphs
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user 2233
A Hamiltonian graph (directed or undirected) is a graph that contains a Hamiltonian cycle, that is, a cycle that visits every vertex exactly once.
3
votes
Accepted
Does this graph contain at least two Hamiltonian cycles?
Here is a proof of Gordon's claim. We will prove something slightly stronger.
Claim.
Let $G$ be a $d$-regular graph with $d$ odd. Then for every $e \in E(G)$, there is an even number of Hamiltoni …
- 32.1k
3
votes
How to efficiently find a Hamiltonian cycle in a graph whose closure is complete?
In the case that your graph satisfies the conditions of Ore's theorem (so that it's Ore closure is $K_n$ after 'one step'), there is an easy algorithm to find a Hamilton cycle.
Arbitrarily arrange …
- 32.1k
9
votes
What is the smallest uniquely hamiltonian graph with minimum degree at least 3?
I am not sure what the smallest such graph is, but since you also asked for more information on uniquely hamiltonian graphs with minimum degree $3$, Entringer and Swart proved the following nice theor …
- 32.1k
4
votes
Accepted
Hamiltonian cycle in $S_n$ with transpositions
Yes, $(S_n, E_n)$ contains a Hamiltonian cycle for every $n \geq 3$. This follows by the Steinhaus–Johnson–Trotter algorithm
. The transpositions can even be chosen to be consecutive elements in the …
- 32.1k
10
votes
Accepted
How many hamiltonian cycles can be removed from a complete directed graph before it becomes ...
I will rephrase your question slightly. Let $K_{n}^{*}$ be the directed graph with $n$ vertices and two oppositely directed edges for each pair of vertices. Your question is then the following.
…
- 32.1k
6
votes
Efficient Hamiltonian cycle algorithms for graph classes
One class of graphs for which many NP-hard problems (including finding a Hamiltonian cycle) are easy (linear-time) are graphs of bounded tree-width. Indeed, by Courcelle's theorem any problem which c …
- 32.1k
5
votes
Accepted
Approximation of Hamiltonian cycles
Claim. For every $\rho \geq 1$, there is no polynomial $\rho$-approximation algorithm for $\texttt{MinHalfSimpCycle}$, unless P=NP.
Proof. Let $G$ be an instance of the Travelling Salesman Problem (TS …
- 32.1k