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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.

15 votes
2 answers
1k views

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 …
18 votes
1 answer
1k views

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 …
6 votes
0 answers
208 views

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 …
46 votes
8 answers
5k views

Can a problem be simultaneously polynomial time and undecidable?

The Robertson-Seymour theorem on graph minors leads to some interesting conundrums. The theorem states that any minor-closed class of graphs can be described by a finite number of excluded minors. As …
34 votes
1 answer
775 views

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 …
8 votes
0 answers
214 views

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 …
14 votes
1 answer
759 views

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 …
9 votes
0 answers
186 views

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 …
7 votes
1 answer
207 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 graph …
6 votes
2 answers
756 views

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 $K_{3,3}$) …
7 votes
3 answers
252 views

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 …
9 votes
0 answers
245 views

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: …
8 votes
0 answers
122 views

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 …
6 votes
0 answers
129 views

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 votes
0 answers
448 views

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 …

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