**9**

votes

**0**answers

104 views

### Cycles of length $2^n - 2$ in the De Bruijn graph

It is well known that the number of (cyclic) De Bruijn sequences is $2^{2^{n-1}-n}$. This number may also be interpreted as the number of cycles of length $2^n$ in the De Bruijn graph of order $n$.
...

**1**

vote

**0**answers

31 views

### Generate connected subgraphs as the satisfying assignments to a SAT instance

I want a SAT instance (in CNF) whose set of satisfying assignments are the connected subgraphs of a given input graph. A general solution would be helpful, but I really only need this when the input ...

**-2**

votes

**1**answer

66 views

### Graph isomorphism for twin free graphs

Suppose you are given two graphs $G_1$ and $G_2$ and are promised that both are twin free. Is the problem of determining if they are isomorphic graph isomorphism hard? I am curious for the cases of ...

**11**

votes

**1**answer

686 views

### Reasons for difficulty of Graph Isomorphism and why Johnson graphs are important?

In http://jeremykun.com/2015/11/12/a-quasipolynomial-time-algorithm-for-graph-isomorphism-the-details/ it is mentioned:
'In discussing Johnson graphs, Babai said they were a source of “unspeakable ...

**3**

votes

**1**answer

203 views

### Is there a version of Robertson-Seymour's graph minor theorem known to apply to signed graphs?

Here a signed graph is one where each edge is signed either odd or even. A cycle is odd or even according to the sum of the signs of its edges. For a given signed graph, a resigning may be performed ...

**3**

votes

**1**answer

108 views

### Minimal number of vertices in a graph with special Hadwiger partitions

Let $G=(V,E)$ be a simple, undirected graph.
We call a partition ${\cal P}$ of a non-empty subset of $V$ a Hadwiger partition if
every block (member of ${\cal P}$) is non-empty and connected, and
...

**-2**

votes

**1**answer

66 views

### Hadwiger partitions where one block is always a singleton

Let $G=(V,E)$ be a simple, undirected graph.
We call a partition ${\cal P}$ of a non-empty subset of $V$ a Hadwiger partition if
every block (member of ${\cal P}$) is non-empty and connected, and
...

**2**

votes

**0**answers

122 views

### Which functions preserve the connectivity of graphs/components?

I am somewhat stuck working on an issue and would really love some guidance. I will state the problem, my current state and what led to it in case the solution lies beyond where I was looking
The ...

**2**

votes

**0**answers

60 views

### Blossoms and Colorings

There is a striking analogy between finding maximum matchings in graphs and determining the chromatic number of graphs: both problems are fairly easy for bipartite graphs, but harder, resp. too hard ...

**2**

votes

**1**answer

234 views

### Can we solve Hamiltonian Path problem for biconnected planar graphs in linear time?

Assume that we have a bi-connected planar graph $G$ with $\Delta(G)>3$, and we want to find a Hamiltonian Path in $G$. As we know the st-order of a bi-connected planar graph can be computed in ...

**2**

votes

**1**answer

49 views

### Bounds on chromatic index

Let $H$ be a hypergraph of maximum vertex-degree $\Delta$. (That is, for all vertices $x$, we have $| \{ e \in H \mid x \in e \} | \leq \Delta$) Are there any bounds on the chromatic index $\chi_e(H)$ ...

**3**

votes

**2**answers

165 views

### on counting the number of trees on Kn (case)

During my reasearch I have stumbled across a problem that can be presented in such way:
"How many are there spanning trees on Kn such that every tree contains v: deg(v) = k, for a given k"
The ...

**2**

votes

**1**answer

138 views

### Extremal combinatorics on bipartite graphs

One open question in extremal graph Theory is the so-called Zarankiewicz problem
(see for instance the wikipedia page), which ask for the maximum number of edges in a bipartite graph with a fixed ...

**8**

votes

**0**answers

313 views

### Is there an “Erlangen Program” for Graph Theory?

There are certain graph theoretic problems (especially optimization problems), whose solution-subgraph (i.e. the set of vertices and edges)), is invariant under certain modifications (especially ...

**1**

vote

**0**answers

35 views

### Atomic parts of lexicographic products of vertex-transitive graphs

Suppose $H_1$ and $H_2$ are connected, vertex-transitive graphs, $H_1$ is not the complete graph, and $|V(H_2)| \ge 2$. Then, the lexigraphic product $G=H_1 \circ H_2$ is vertex-transitive, $0 < ...

**1**

vote

**3**answers

135 views

### In what types of graphs can the maximum independent set be found in polynomial time?

I need to find the maximum independent sets of a serial of regular graphs, which is generally NP-complete.
The wikipedia told me that this problem can be solved in polynomial time if the graph is ...

**1**

vote

**1**answer

62 views

### Reduced echelon form of sparce matrices and constructing hash function

Let $G$ be a $d$-regular graph, and $A$ be the incidence matrix of $G$. Also suppose $B$ is a reduced echelon form of $A$ such that computations are in $\mathbb F_2$. Given matrix $B$, can we find ...

**0**

votes

**0**answers

14 views

### Two-optimality of the Union of a Shortest Hamilton Cycle and a Minimum-weight Maximal Matching

let $G(V,E)$ be a complete, finite, symmetric and simple weighted graph with a unique shortest Hamilton cycle $T_{opt}(G)$ and a unique maximum matching of minimal weight $M_{opt}(G)$.
Is it ...

**3**

votes

**0**answers

129 views

### Finiteness for 2-dimensional contractible complexes

While thinking about graph-complex and related operadic stuff, I found a quite interesting (at least for me) question. However, I'm a novice in the algebraic topology, so I'm unable to resolve it by ...

**5**

votes

**1**answer

148 views

### Generalisation of Kuratowski

So I've recently read the infinite graph version of Kuratowski's theorem. It says that a graph $G$ is planar if and only if the following three conditions holds:
$|V(G)| \le |\mathbb{R}|$
$G$ has at ...

**1**

vote

**1**answer

89 views

### Assigning random orientation to an edge in a regular graph

Given a simple regular graph of degree $d$ on $n$ vertices.
Assume an ordering of vertices and assume all orientations of edges is from $i$ to $j$ if edges $ij$ exists and $i<j$. Pick $m$ random ...

**-1**

votes

**1**answer

106 views

### Reducing chromatic number

(1) Is there an estimate for maximum number of edges in a $k$ colorable $v$ vertex $d$ degree graph with genus $g$? Call this $|E|$?
(2a) What is a good estimate for worst case number of edges that ...

**2**

votes

**1**answer

74 views

### Do product distributions (or graph products) eventually cluster as more products are taken?

Say we have a joint distribution on a finite alphabet $\mathcal{X}\times \mathcal{Y}$. It could be a communication link where we want to send a random message $X$ over a channel, but it gets garbled ...

**2**

votes

**1**answer

213 views

### Proof that it's possible to colour all elements in set, that all subsets will be bicolored

(For my easy understanding, let me rewrite the question. The author should feel free to remove my edit or... accept it; I am leaving the original formulation at the end intact).
=================
...

**1**

vote

**1**answer

142 views

### Finding many disjoint sub-trees with many leaves

Let $T$ be a rooted binary tree with $L$ leaves, and let $\ell$ be a natural number smaller than $L$. The question is what is the maximal number of disjoint rooted sub-trees with at least $\ell$ ...

**2**

votes

**1**answer

83 views

### “Hypo” and “Hyper” for Perfect Matching

There is a fairly rich classification on graphs with respect to the existence of Hamiltonian cycles either in unmodified graphs or after certain small modifications.
Do there also exist such ...

**8**

votes

**2**answers

373 views

### Densest Graphs with Unique Perfect Matching

Given a graph $G$ with $n$ vertices, that has a perfect matching $M$, what is the maximal number of edges that $G$ can have without contradicting the uniqueness of $M$?
Are examples of such extremal ...

**1**

vote

**1**answer

42 views

### Test Instances for Perfect Matchings in Graphs

Are there any graphs with a known set of perfect matchings and other predefined properties, such as vertex connectivity, which can be used for testing the implementation of matching algorithms?
...

**0**

votes

**0**answers

51 views

### What is wrong with this isomorphism preserving transformation to a graph of bounded clique width and bounded rank width?

Got an isomorphism preserving transformation to a graph of bounded
clique width and rank width. It, a paper and graphclasses.org imply
graph isomorphism is in P, so likely something is wrong.
Let $G$ ...

**4**

votes

**3**answers

130 views

### Repository of graph classes that are tough to test non-isomorphic pairs from isomorphic pairs

(1) Which graph classes are extremely tough to test for graph non-isomorphic pairs from isomorphic pairs?
(2) Is there a repository of adjacencies from such classes?

**3**

votes

**2**answers

132 views

### Database of adjacency matrices on cospectral non-isomorphic graph pairs

Is there a repository of cospectral non-isomorphic graphs available somewhere?
I am looking for list of $0/1$ adjacency matrix pairs that can be input data in tools such as MATLAB.

**7**

votes

**4**answers

260 views

### Always a planar-drawn cycle through $n$ points

Given $n$ points in the plane, can we always find a cycle through all of them that has only straight line edges and no edges intersect (planar-drawn)?
Intuitively the answer is yes, but I am ...

**1**

vote

**0**answers

18 views

### Path Sums in Arc Labeled st-graphs

My research has led me to a question on sums of integer labeled arc paths in an st-graph (single source single sink, acyclic, I actually have multi-graphs). The problem is to label all the arcs in a ...

**7**

votes

**0**answers

77 views

### A separation property of graphs of bounded tree-width

The following separation property of trees is well-known and in fact easy to prove (see e.g. the paper "Covering a hypergraph of subgraphs" by Noga Alon, Lemma 2.2)
Let $T$ be a tree and $r, m$ ...

**0**

votes

**0**answers

20 views

### A Statement about a General Property of Negative Cycle Detection Algorithms

in this paper from 1999, the authors Boris Cherkassky and Andrew Goldberg state in the abstract that
"The negative cycle problem is to find a negative length cycle in a network or to prove that ...

**0**

votes

**1**answer

50 views

### Cycle-intersecting subsets

Let $G=(V,E)$ be a finite, simple, undirected graph. We call $D\subseteq V$ cycle-intersecting if for every simple cycle $C\subseteq V$ we have $C\cap D \neq \emptyset$.
Is there a graph $G$ such ...

**0**

votes

**1**answer

58 views

### What is the definition of size of an edge$?$

In page-$13$ of Graph minors. $X$. Obstructions to tree-decomposition, $\gamma(G)$ introduced as maximum size of an edge. What is the definition of size of an edge$?$ I think it may be number of edges ...

**3**

votes

**1**answer

59 views

### What is the relation between Treewidth and Order of graph?

Can we say in general, Treewidth of every graph is less than or equal to number of vertices in that graph?
Is there any other general relation between Treewidth and Order of graphs?

**2**

votes

**1**answer

48 views

### What is the relation between Hadwiger number and Treewidth?

Is there any general relation between Hadwiger number and Treewidth of a graph? Intuitively I think Hadwiger number is greater than or equal to Treewidth, but I couldn't prove it.

**4**

votes

**1**answer

121 views

### Which graphs are prime under the Cartesian product?

I'm looking for a characterization of graphs that are prime under the Cartesian product, with prime defined as in this question. Does such a characterization exist, either in general or after ...

**4**

votes

**1**answer

126 views

### does every vertex-cut set in a maximal planar graph contain a cycle?

$G = (V, E)$ is a 3-connected plane triangulation. Let $S \subset V$ such that $G(V - S)$ is disconnected. Is it true that $G(S)$ must contains a separating cycle?
My intuition is leading me to ...

**1**

vote

**1**answer

73 views

### What is the relation between size of maximum clique and branchwidth?

Let $bw(G)$ be the branchwidth of graph $G$ and $\omega(G)$ be the size of maximum clique in $G$. I think the following inequality holds:
$$
\omega(G)\leq bw(G)
$$
Intuition: Assume (in reverse of ...

**0**

votes

**0**answers

42 views

### Arithmetic progressions on a graph

Given $K_k$ a complete graph what is the minimum $n\in\Bbb N$ needed so that there is a map:
$$f:\{0,1,\dots,n\}\rightarrow\mathsf{Edges}(K_k)$$ which makes every simple cycle to be $r$-term ...

**0**

votes

**0**answers

62 views

### Almost-hypohamiltonian Kneser Graphs

Here, it states that:
When $n\ge 3k$, the Kneser graph $KG_{n,k}$ always contains a Hamiltonian cycle. Computational searches have found that all connected Kneser graphs for $n \le 27,$ except for ...

**1**

vote

**0**answers

51 views

### How effective is using local property to test Shannon capacity?

A key tool in graph theory is the laplacian which is a local property. We can form a semidefinite programming and get an upper bound for Shannon capacity using laplacian.
Shannon capacity is ...

**2**

votes

**1**answer

78 views

### Repeated nodes in tree-decomposition of a graph is allowed or not?

As we know, a tree-decomposition of a graph must have following features:
All vertices are covered
All edges are covered
The connectivity condition
I think using repeated nodes in ...

**1**

vote

**1**answer

103 views

### $q$-connectedness of random digraphs obtained from a fixed graph

Let $G = (E,V)$ be an undirected graph (which can have multiple edges or loops).
Let $k,l,m\colon E\to \mathbb{R}_{\geq 0}$ be three edge-weight functions that satisfy $2k(e) + l(e) + m(e) = 1$ for ...

**-2**

votes

**1**answer

156 views

### Planar Graphs with #Vertices = #Faces [closed]

Do you know anything special about that kind of planar graphs? An article that covers these graphs might be helpful.

**3**

votes

**0**answers

104 views

### Construction of algebraic curves using line bundles on graphs

In this paper http://arxiv.org/abs/0707.1309 Matthew Baker and Serguei Norine, construct a analogue of the Riemman Roch formula for Lineal Systems defined on graphs. In the paper ...

**1**

vote

**1**answer

181 views

### On Knot Equivalence problem statement

How is the knot equivalence problem represented?
By this I mean I am looking for an analogy that compares with graph equivalence. For graph equivalence, we have two graphs $G_1$ and $G_2$ with ...