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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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1answer
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### The number of Hamiltonian circuits on a convex polytope embedded in $\mathbb{R}^N$

Recently I wondered whether there might be a natural topological complexity measure for convex polytopes embedded in $\mathbb{R}^N$. After some reflection it occurred to me that the number of distinct ...
0answers
47 views

### Normal colorings of bridgeless cubic graphs

Definition (informal) A normal edge-5-coloring of a bridgeless cubic graph $G$ is a proper 5 coloring of the edges of the graph, so that for each edge $e\in E(G)$, either $e$ and the four edges ...
3answers
404 views

### Arranging all permutations on $\{1,\ldots,n\}$ such that there are no common points

If $n>0$ is an integer, let $[n]=\{1,\ldots,n\}$. Let $S_n$ denote the set of all permutations (bijections) $\pi:[n]\to[n]$. For which positive integers $n$ is there a bijection $\Phi:[n!]\to S_n$ ...
1answer
106 views

### Edge colorability and Hamiltonicity of certain classes of cubic graphs (MO graphs)

Let $G$ be a simple cubic graph (that is, 3-regular). A dominating circuit of $G$ is a circuit $C$ such that each edge of $G$ has an endvertex in $C$. The circuit $C$ is chordless if no edge which is ...
1answer
151 views

### Which are good algorithms for finding Hamiltonian path (not necessarily a circle) up to now?

I am not expertise in graph theory. So have to ask this question here. The term "good" means that the algorithms should be efficient for general undirected simple connected graphs with a higher ...
1answer
743 views

### Are all cubic graphs almost Hamiltonian?

Call a graph $G$ $n$-almost-Hamiltonian if there is a closed walk in $G$ that visits every vertex of $G$ exactly $n$-times. So a Hamiltonian graph is $n$-almost-Hamiltonian for all $n$. Are all ...
0answers
133 views

### Is there a permutation $\pi\in S_n$ with $\sum\limits_{0<k<n}\frac1{\pi(k)^2-\pi(k+1)^2}=0$ for each $n>7$?

Let $S_n$ be the symmetric group of all permutations of $\{1,\ldots,n\}$. QUESTION: Is it true that for each $n=8,9,\ldots$ we have $$\sum_{0<k<n}\frac1{\pi(k)^2-\pi(k+1)^2}=0\tag{*}$$ for ...
0answers
158 views

### Quantitatively characterizing the failure of the converse of Dirac's theorem

First, I am an undergraduate so I apologize if this is trivial and certainly understand if it is closed immediately. I am currently in a combinatorics and graph theory class and recently we have ...
1answer
140 views

### What is the complexity of counting Hamiltonian cycles of a graph?

Since deciding whether a graph contains a Hamiltonian cycle is $NP$-complete, the counting problem which counts the number of such cycles of a graph is $NP$-hard. Is it also $PP$-hard in the sense ...
1answer
155 views

### Grinberg's uniquely hamiltonian 3-connected graphs (Russian paper)

Many years ago, Grinberg found some uniquely-hamiltonian $3$-connected graphs, and published his results in a paper that has been cited several times as follows. E. Grinberg, Three-connected graphs ...
0answers
112 views

### Halin Graphs with Highest Number of Hamilton Cycles

Halin graphs contain a Hamilton cycle and have the interesting property, that, also in the case of arbitrary real edge weights, it is possible to report one of the shortest contained Hamilton cycles ...
0answers
69 views

### Reference request: Bipartite symmetric graphs are hamiltonian

Does anyone know whether bipartite symmetric graphs are hamiltonian? I'm not sure whether anyone have proved it before, but a nonhamiltonian symmetric bipartite graph would lead to a counterexample to ...
1answer
109 views

### Why is the number of Hamiltonian Cycles of n-octahedron equivalent to the number of Perfect Matching in specific family of Graphs?

In OEIS A003436, it is written that the number of inequivalent labeled Hamilton Cycles of an n-dimesnional Octahedron is the same as the number of Perfect Matchings in a the complement of the Cycle ...
1answer
327 views

### Are bipartite Moore graphs Hamiltonian?

This is motivated by a computer-generated conjecture that bipartite distance-regular graphs are hamiltonian. I decided to check the case of Moore graphs first. The cycles and complete bipartite ...
0answers
36 views

### Maximal non-hamiltonian graphs - spanned by a theta graph?

At the moment I am interested in maximal non-hamiltonian graphs, so that is a (simple, undirected) graph that does not itself have a hamilton cycle, but if you add an edge between any two distinct non-...

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