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Results tagged with graph-theory
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user 1492
Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.
6
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Maximum number of Hamilton paths in a tournament on $n$ vertices
Recall that a tournament is a directed graph $T$ such that for every pair of distinct vertices $\{v,w\}$, exactly one of the ordered pairs $(v,w)$, $(w,v)$ is an arc of $T$.
A tournament is strongly c …
- 12.7k
34
votes
1
answer
775
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Which graphs on $n$ vertices have the largest determinant?
This is a question that seems like it should have been studied before, but for some reason I cannot find much at all about it, and so I am asking for any pointers / references etc.
The determinant of …
- 12.7k
8
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0
answers
214
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Has anyone implemented a circle graph recognition algorithm?
A double occurrence word is a circular string of length $2n$ over an alphabet of size $n$ with each letter occurring exactly twice, for example:
ABACCDBD
Given a double occurrence word, we can form …
- 12.7k
7
votes
1
answer
207
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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 graph …
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9
votes
0
answers
186
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Partitioning the vertices of a graph into induced trees
I am looking for previous work regarding graphs whose vertices can be assigned colours (not necessarily a proper colouring) in such a way that each colour class induces a tree.
In particular I am mos …
- 12.7k
7
votes
3
answers
252
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How many $40$-vertex cubic bipartite graphs have determinant $\pm 3$?
To get some feel for the size of a particular computation, I would like to know the approximate number of (pairwise-nonisomorphic) cubic bipartite graphs on $40$ vertices whose bipartite adjacency mat …
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14
votes
1
answer
759
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What was Smith's proof of Smith's theorem on Hamilton cycles in cubic graphs?
In a short 1946 paper "On Hamiltonian Circuits", Tutte proved the famous result that an edge in a cubic graph lies in an even number of Hamilton circuits.
He attributed the result to his friend CAB S …
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9
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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: …
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8
votes
0
answers
122
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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
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 triangulati …
- 12.7k
15
votes
2
answers
1k
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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 pas …
- 12.7k
22
votes
0
answers
546
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Zero curves of Tutte Polynomials?
There is an extensive theory of the real and complex roots of the chromatic polynomial of a graph, a substantial fraction of this being due to the connections between the chromatic polynomial and a pa …
- 12.7k
5
votes
0
answers
114
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Implementations of Tutte polynomial [reference request, of a kind]
This question is not a 100% fit for MO, but it is a serious question that can be viewed as a sort of reference request, and I think fits here more than elsewhere.
I have been asked to write a chapter …
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12
votes
0
answers
448
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Colouring a graph whose edge set is a special union of cliques
I am trying to show that a certain family of graphs can always be properly coloured with at most $6$ colours (where "properly coloured" means that each vertex gets a colour and no edge has both ends t …
- 12.7k
18
votes
1
answer
1k
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Bicycles and spanning trees of graphs
A spanning tree in a graph is a connected spanning subgraph with no cycles; it is well known that the number of spanning trees can be found by taking the determinant of a certain matrix related to the …
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