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Results tagged with graph-theory
Search options questions only
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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.
8
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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
10
votes
2
answers
2k
views
Graphs where every two vertices have odd number of mutual neighbours
There was a rather cute question last week about graphs where every pair of distinct vertices has an odd number of mutual neighbours.
The question was to show that such a graph must have an odd numbe …
- 12.7k
8
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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
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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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 …
- 12.7k
5
votes
2
answers
665
views
Complexity of determining if two graphs have same cycle matroid?
Consider the following question:
Input: Two graphs G1 and G2
Question: Is the cycle matroid M(G1) isomorphic to the cycle matroid M(G2)
What is the complexity of this question?
It is well known th …
- 12.7k
9
votes
1
answer
480
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Has anyone seen this sort of graph property used before?
Consider the following property of a graph $G$:
The graph $G$ has no independent cutset of vertices, $S$, such that the number of components of $G-S$ is more than $|S|$ (the size of $S$).
(That is, …
- 12.7k
6
votes
5
answers
3k
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Where is it shown how to construct a decomposition tree for a series-parallel graph in linea...
There are two common ways to define a series-parallel graph (or 2-terminal series-parallel graph).
Definition 1
start with K_2 marking both vertices as terminals
repeatedly join two smaller 2-termi …
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12
votes
1
answer
2k
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4-regular graphs with every edge in a triangle
I am interested in regular graphs in which every edge lies in a triangle.
For 3-regular graphs, only the complete graph $K_4$ has this property, so there's not much to see here.
For 4-regular graphs …
- 12.7k
11
votes
1
answer
914
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Uniquely hamiltonian graphs with minimum degree 4
A graph is uniquely hamiltonian if it has exactly one Hamilton cycle.
As every edge in a cubic graph lies in an even number of Hamilton cycles, a cubic graph cannot be uniquely hamiltonian, and a fa …
- 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
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 …
- 12.7k
6
votes
2
answers
756
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Minor-closed classes of graphs with large numbers of excluded minors
Robertson and Seymour tell us that any minor-closed family of graphs has a finite collection of excluded minors.
Standard examples include planar graphs with two excluded minors ($K_5$ and $K_{3,3}$) …
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9
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0
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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
6
votes
0
answers
208
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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 …
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