All Questions
Tagged with hamiltonian-graphs graph-theory
37 questions with no upvoted or accepted answers
12
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1
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Hobbled rook tour – Hamiltonian cycle on square grid
Consider a square grid of even side length ($2n \times 2n$). It is easy to see that there must exist a Hamiltonian cycle on the corresponding grid graph. Such a cycle is called balanced if the number ...
- 121
9
votes
0
answers
245
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Heuristic arguments regarding Sheehan's conjecture?
Sheehan conjectured that there are no 4-regular graphs that are uniquely hamiltonian (i.e. have exactly one hamilton cycle).
Evidence that might be loosely seen to be in favour of this conjecture is: ...
- 12.7k
8
votes
0
answers
123
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Is there a non-bipartite hamiltonian cubic graph on $n$ vertices with no $(n-1)$-cycle?
Is there a cubic (3-regular) graph $G$ on $n$ vertices such that:
$G$ is hamiltonian
$G$ has no $(n-1)$-cycles
$G$ is not bipartite
My computer tells me that there are none on up to $24$ vertices.
- 12.7k
6
votes
0
answers
164
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Hamilton cycles in random graphs with just enough connectivity
What is the asymptotic probability that $G$ has a Hamilton cycle if $G$ is a random $n$ vertex $\frac{4}{3}n$ edge graph, with minimum degree 2 and without degree 2 vertices at distance 1 or 2 to each ...
- 7,545
6
votes
0
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73
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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 ...
- 1,018
6
votes
0
answers
129
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Minimum number of hamilton cycles in a 4-connected planar triangulation?
I am currently interested in hamilton cycles (i.e. a cycle through every vertex) in planar triangulations (i.e. planar graphs with every face a triangle).
There are non-hamiltonian planar ...
- 12.7k
6
votes
0
answers
76
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Cage graphs and even cycles
Let $G$ be a $(\nu,g)$-cage graph of degree $\nu$ with girth $g$ and $n=n(\nu,g)$ vertices.
Based on the known examples, I am wondering if the following can be proved/disproved:
Is it true that ...
- 13.4k
6
votes
0
answers
108
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Localizing Bondy's metaconjecture on hamiltonicity
Definitions:
Let $G$ be a graph on $n$ vertices. $G$ is Hamiltonian provided $G$ has a cycle of length $n$. $G$ is pancyclic provided $G$ has a cycle of length $\ell$ for every $3 \leq \ell \leq n$.
...
- 399
5
votes
0
answers
127
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Do uniquely Hamiltonian graphs have cycles of a sufficiently long length?
Let $C$ be a Hamiltonian cycle of a graph $G$.
Call an edge $e$ of $G$ a chord if $e\not\in C$.
Let each edge of $C$ be weighted $1$ and each chord be weighted $2$.
The weight of a path or cycle of ...
- 673
5
votes
0
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154
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How to construct 4-regular graphs with few Hamiltonian decompositions?
A Hamiltonian decomposition of a finite simple graph is a partition of its edge set so that each partition class forms a Hamiltonian cycle. This is only possible if the graph is $2k$-regular.
...
- 13.6k
5
votes
0
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99
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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 ...
- 25.4k
5
votes
0
answers
295
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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 ...
- 17.6k
4
votes
0
answers
69
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is a 4-connected planar graph still Hamiltonian after removing an edge?
We know that 4-connected planar graphs are Hamiltonian(by the known Tutte Theorem). Additionally, Thomas and Yu [1] proved that removing two vertices from a 4-connected planar graph still preserves ...
- 1,851
4
votes
0
answers
234
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How many 20-vertex 2-connected 5-regular non-Hamiltonian graphs are there?
As for the question in title, I attempted to use nauty to obtain them, but it has been running on my computer for nearly three days without producing any results.
<...
- 1,851
4
votes
0
answers
230
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Is this case of Barnette's Conjecture known?
Context: Barnette's Conjecture is that every bipartite cubic polyhedral graph is Hamiltonian. I have been interested by this problem for a long time, and I recently came up with a result. From my ...
- 3,499
4
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0
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143
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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 ...
- 13.2k
4
votes
0
answers
258
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When is an induced subgraph of a Johnson graph hamilton-connected?
The Johnson graph $J(n,k)$ has as its vertices the $k$-subsets of $\{1, 2, \dots, n\}$ where two vertices are adjacent iff their intersection has size $k-1$. A graph is Hamilton-connected if every two ...
- 393
3
votes
0
answers
70
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Hamilton cycles in Cayley graphs: between Rapaport-Strasser and Fleischner
A well-known question of Rapaport-Strasser asks whether every finite connected Cayley graph has a Hamilton cycle. Fleischner's Theorem implies that if $S$ is the generating set of such a Cayley graph $...
- 1,936
3
votes
0
answers
108
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Hamiltonian path in $\{0,1\}^n$ with rotations and bit-flip in position 0
We consider any non-negative integer as an ordinal, that is $0=\emptyset$ and $n=\{0,\ldots,n-1\}$ for every positive integer. Let $\{0,1\}^n$ denote the set of $\{0,1\}$-vectors of length $n$.
Define ...
- 48.9k
3
votes
0
answers
76
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Hamiltonian cycles in Cayley graph on alternating group
Let $G=\operatorname{Cay}(A_n,S)$ be the Cayley graph on the Alternating group $A_n\quad n\ge4$ with generating set $S=\{(1,2,3),(1,2,4),\ldots,(1,4,2),(1,3,2)\}$. One Hamiltonian cycle in $G$ for $n=...
- 2,089
3
votes
0
answers
107
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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 ...
- 9,501
3
votes
0
answers
145
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Hamiltonian cycle polytope for the hypercube graph
Let $Q_n$ denote the $n$ dimensional hypercube graph (i.e., graph formed from the vertices and edges of an n-dimensional hypercube). Denote the set of edges and vertices of $Q_n$ by $E_n$ and $V_n$ ...
- 393
3
votes
0
answers
129
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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 $C$.
...
- 6,962
2
votes
0
answers
112
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Constructing Hamiltonian circuits in acyclic digraphs
Any directed graph $G$ lacking cycles can acquire a Hamiltonian circuit through the addition of a sufficient number of edges.
Q. Is there a method to minimize the addition of edges to achieve a ...
- 4,058
2
votes
0
answers
116
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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
...
- 25.4k
2
votes
0
answers
184
views
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 ...
- 1,005
1
vote
0
answers
65
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Has there been progress on Hamiltonicity in 4-connected claw-free graphs with a constant maximum degree?
In 1984, Matthews and Sumner [1] conjectured that every 4-connected claw-free graph is Hamiltonian, and this conjecture is still wide open.
I would like to know if there has been any progress on this ...
- 1,851
1
vote
0
answers
93
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15-game graph contains a Hamiltonian path ? Lovász conjecture for groupoids, loops, quasigroups , etc?
Typically Cayley graphs are defined for groups and generators sets S. But basically one only needs some set S and another set V and partially defined operation SxV->V, then one defines graph with ...
- 24.9k
1
vote
0
answers
61
views
A generalized/set hamiltonian cycle problem on directed graphs
So this problem originally stems from the asymmetric generalized/set TSP problem, where I am interested in asking the question which or how many edges I can delete while maintaining feasability. The ...
- 111
1
vote
0
answers
147
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Properties of graphs with Hankel-like adjacency matrix
I am having undirected graphs with adjacency matrices which have a regular Hankel-like form, e.g.,
$$A=\begin{pmatrix}0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & (6\times 0 \text{ ...
- 243
0
votes
0
answers
86
views
Uniqueness of compatible cycle decomposition for Eulerian trail
Fleischner mentions in his article Uniqueness of maximal dominating cycles in 3-regular graphs and of hamiltonian cycles in 4-regular graphs about the uniqueness of compatible cycle decomposition that ...
0
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0
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36
views
A class of directed graph, when their minimal polynomial of the adjacency matrix matches the characteristic polynomial
We consider an unweighted directed simple graph, $G$, with a Hamiltonian cycle.
Q. Assume that the adjacency matrix of $G$ is non-singular. Do the characteristic and minimal polynomials of the ...
- 4,058
0
votes
0
answers
52
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Are there 4-connected planar non-hamilton multi-graphs?
Tutte proved the famous result: Every planar 4-connected graph has a hamiltonian cycle. But I read in Section 111.6.5 on book Eulerian Graphs and Related Topics that the author Herbert Fleischner ...
- 1,851
0
votes
0
answers
62
views
Clique sizes of generalized Kneser graphs
Are there known bounds for clique size in generalized Kneser graphs $KG(n,k,t)=K(n,k,t-1)$, the graph formed by distinct $k$ subsets of $n$ set so that two subsets with at most $t$ elements in common ...
- 2,089
0
votes
0
answers
65
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Cycles in Kneser graphs with three vertices forming triangles
Consider the Kneser graphs $G=K(n,k)$. Is it possible to list how many even cycles, or, at the least, existence of an even cycle of a given order in $G$, such that any three consecutive vertices form ...
- 2,089
0
votes
0
answers
125
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Are there any necessary conditions of the existence of a Hamiltonian cycle on directed graphs
I'm trying to prove that one concrete directed graph has no Hamiltonian cycle, but didn't seem to find any relevant theorems
0
votes
0
answers
64
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Degree-constraints for the existence of vertex-disjoint directed cycle covers in digraphs
Given a digraph $G(E,V): (u,v)\in E\implies(v,u)\notin E$, what is known about lower bounds on the indegree and outdegree of the vertices that guarantee the existence of a vertex-disjoint directed ...
- 13.2k