**0**

votes

**0**answers

38 views

### Cycle cover of a cubic polyhedral graph

Tutte graph is a counterexample for the Tait's conjecture stating that all cubic graphs are Hamiltonian. For the non-hamiltonian graphs - is it true that all vertices of any such graph can be covered ...

**2**

votes

**0**answers

48 views

### Regular graphs with unimodal subdegrees that are not distance-regular

Distance regular graphs are known to exhibit the following property: starting from an arbitrary vertex $\alpha$, let $k_i$ denote the number of vertices at distance $i$ from $\alpha$ (in terms of ...

**1**

vote

**0**answers

68 views

### labeling vertices in a graph to minimize distance between adjacent vertices

We have a regular graph $G$ of degree $m$ on $n$ vertices and we label each of its vertices with the number $1$ through $n$. What can we say about the maximum difference between the numbers of two ...

**7**

votes

**2**answers

273 views

### Wait time to grid network disconnection with failing edges

Let $G_n$ be an $n \times n$ planar toroidal grid graph, with each node
connected to its four neighbors, with the top row connected to the bottom,
and the right column connected to the left.
Suppose ...

**2**

votes

**0**answers

61 views

### Characterize the equivalence class of bipartite graphs obtained from each other by elementary row operations on their adjacency matrices

Let $M$ be an $m\times n$ matrix real matrix. Let $G$ be a bipartite graph, with partitions $A$ and $B$, such that $|A|=m$ and $|B|=n$. A node $i\in A$ is linked to a node $j\in B$ if and only if ...

**7**

votes

**1**answer

356 views

### Paths in groups

Given a finite group $G$, write $K(G)$ for the complete digraph on the elements of $G$. Label the edge from $g$ to $h$ by element $g^{-1}h$.
Question: For what groups does there exist a Hamiltonian ...

**1**

vote

**1**answer

128 views

### Sequences that represent different drawing of chords?

In combinatorics there are there are special kind of sequences, in which their terms represent the number of different ways that we can draw chords with some properties.
Actually my question is ...

**0**

votes

**0**answers

92 views

### Has this type of Cayley graphs been studied before?

Let $\mathbb F_q $ be a finite field of characteristic 2.
Let $ x^2 + Sx +P \in \mathbb F_q[x] $ be an irreducible polynomial over $ \mathbb F_q $.
Let $$ T = \{ \left( \begin{matrix}
...

**17**

votes

**0**answers

156 views

### A small unavoidable collection of subgraphs

What is the smallest number S(k,n) of unlabeled graphs on k vertices such that every simple graph on n vertices contains at least one of these as an induced subgraph?
I'd like to avoid exhaustive ...

**4**

votes

**3**answers

147 views

### Is there any partition of a regular graph which in any part there exists a vertex with all its neighborhood?

Let we have a regular graph. I want to know if we can partition the vertex set of this graph while in any part there exist a vertex with all its neighborhood?

**10**

votes

**1**answer

278 views

### doubly-stochastic isomorphisms of graphs

A doubly stochastic matrix that commutes with the adjacency matrix of a graph is a doubly-stochastic automorphism of that graph (definition by Tinhofer 1986). Each (classical) automorphism of a graph ...

**2**

votes

**1**answer

79 views

### a necessary condition for a nonempty graph being a line graph

Every gragh below will refer to a finite simple graph.
There is a necessary and sufficient condition for a nonempty gragh being a line gragh:
Krausz's Theorem
A nonempty gragh is a line gragh if ...

**1**

vote

**0**answers

116 views

### Relationship betwen eigenvectors

Suppose that we have two matrices A and B. Matrix B is taken from A with one row and one column deleted. On the other hand A is n*n matrix and B is (n-1)*(n-1) matrix and is created by deleting last ...

**1**

vote

**1**answer

113 views

### Cycles in directed graphs

Let G be a finite directed graph (allowing multiple edges). We define a cycle (as usual) to be a sequence of edges $e_0, e_1, \dots, e_{n-1}$ (up to cyclic permutation) such that the terminal vertex ...

**10**

votes

**1**answer

568 views

### Where can I find Gonthier's Coq code proving the four color theorem?

In a 2008 article in the Notices, Georges Gonthier announced a computer-checked proof of the four color theorem using Coq:
Gonthier, Georges. Formal proof—the four-color theorem.
Notices Amer. ...

**6**

votes

**2**answers

120 views

### How to efficiently find a Hamiltonian cycle in a graph whose closure is complete?

A graph whose closure is the complete graph is Hamiltonian by the Bondy-Chvátal theorem, but I haven't found a polynomial algorithm for finding a Hamiltonian cycle in such a graph. Is there one that ...

**0**

votes

**0**answers

125 views

### Basis of periodic tiling of Wang tile

Given a set of Wang tile,
Given 3 periodic tiling: A, B, C
We define 3 vector F[A], F[B], F[C]
each vector correspond to the appearing frequency of each type of tiles in the tiling.
Now, we ...

**1**

vote

**1**answer

90 views

### simple cycle analog in 2D (with application in tiling)

We know that any closed cycle of a graph could be decomposed into sum of simple cycles. To translate this theorem into tiling of 1D (Wang tile). We know that any 1D periodic tiling could be ...

**3**

votes

**1**answer

119 views

### Cayley graph which is isomorphic to the line graph of a complete graph

From the literature, we know that the line graph of a complete graph $L(K_{q})$ is a Cayley graph if and only if $q \equiv 3$( mod 4) is a prime power. Now, if $q \equiv 3$( mod 4) is a prime power, ...

**-4**

votes

**2**answers

82 views

### About planar graphs? [closed]

Can any non-planar graph with n minimum crossing points be 'drawn' on a sphere so the vertice and edge sets are the same and it has a connected subset A with minimum r crossing points and a disjoint ...

**1**

vote

**0**answers

106 views

### Graph theoretical representation of Wang Tile

We note that for one dimensional tiling problem of Wang Tile could be represented by a graph. Each cycle on the graph represents a periodic solution.
However, is there a well established counter-part ...

**1**

vote

**0**answers

84 views

### What is the projective dual of a planar graph?

Everybody learns the usual definition of the dual of a planar graph when edges are preserved and faces are mapped to vertices. Everybody learns the projective duality. What if we apply it to a ...

**3**

votes

**0**answers

56 views

### Maximal $k$-chordal subgraph

Recall that a graph is called $k$-chordal if any cycle $C$ of length $> k$ contains a chord, i.e. an edge joining to non-consecutive vertices in $C$. Let $f(n, k)$ be the minimal number of edges ...

**1**

vote

**1**answer

95 views

### How to calculate the maximum number of rainbows for arbitrary graphs?

This question is inspired by problem 1 of the combinatorics test of the 2012 third round iranian olympiad which is as follows:
We've colored edges of $K_n$ with $n-1$ colors. We call a vertex rainbow ...

**3**

votes

**2**answers

185 views

### Distance between two networks

Suppose you have networks A and B, each with a set of nodes and edges. You want to measure how similar the networks are to each-other. None of the nodes or edges are labelled. What are the metric(s) ...

**3**

votes

**0**answers

79 views

### Reconstructing a function from its variants that negate one argument

Call two functions $g(x_1,\ldots,x_n)$ and $h(x_1,\ldots,x_n)$ from complex numbers to complex numbers equivalent if they are the same up to the order of their arguments. Formally: there is a ...

**6**

votes

**1**answer

159 views

### Integral straight-line embeddings of planar graphs

Wikipedia says (in the article on Fáry's theorem),
"Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers. The ...

**4**

votes

**1**answer

202 views

### Extremal graph theory for directed graphs

In extremal graph theory, there are results such as
$$t(C_4,G)\geq t(K_2,G)^4,$$
where $G$ is an undirected graph, $C_4$ is a cycle graph on 4 nodes, $K_2$ is a complete graph of $2$ nodes, and ...

**2**

votes

**1**answer

82 views

### Eigenvalues of a graph and its one-edge-delation graph

Let $G$ be any graph with at least one edge and let $e$ be any edge of $G$. Let $G-e$ denote the subgraph of $G$ obtained by deletion of the edge $e$. Assume that $G$ has $n$ vertices.
Suppose ...

**3**

votes

**1**answer

111 views

### A problem related to routing in a graph

I have come across a new problem - I want to know whether this problem is similar to some existing problem or not.
The new problem is this. There is a tourist who has a having the following ...

**0**

votes

**1**answer

167 views

### Spectral Graph Theory [closed]

Let G be an undirected graph, then
Laplacian Matrix(L(G)) = Degree Matrix (D(G)) - Adjacency Matrix (A(G)).
What is the relationship between laplacian and adjacency spectrum of undirected graphs?

**0**

votes

**1**answer

112 views

### Laplacian matrix of a graph with negative weights

I am trying to calculate the Laplacian and Adjacency matrix of a graphs for positive and negative weights. If a graph be simple with only non-negative weight it is easier. But in my graph I have some ...

**7**

votes

**4**answers

315 views

### Extremal examples for a folklore lemma on subgraphs of large minimum degree

It's a well known fact that a graph $G$ of average degree $d$ has a subgraph $G'$ of minimum degree at least $d/2$ and that the constant $1/2$ cannot be improved. The proof I know, which proceeds by ...

**3**

votes

**0**answers

140 views

### Must distinct tree eigenvalues be relatively far apart?

How close to each other can two distinct eigenvalues of a tree be, as a function of the number $n$ of nodes ?
For example, the path $P_n$ exhibits a gap of order $\frac{2\pi^2}{n^2}$ asymptotically ...

**1**

vote

**0**answers

45 views

### Euclidean embedding of a graph based on 1-ring neighborhood distances only

Consider a graph $(V,E)$, $\vert V \vert = n$ and weights $\{l_{ij}\}$, where $l_{ij}>0$ iff there is an edge connecting vertices $v_i$ and $v_j$. Distances beyond the 1-ring neighborhood are not ...

**4**

votes

**1**answer

140 views

### Graph presentation of Lexicographic shifts

Consider a finite alphabet $\{0,1, \ldots, n-1\}$. Let $\Sigma_n = \mathop{\prod}\limits_{j=1}^{\infty}\{0, \ldots n-1\}$ be the set of infinite one sided sequences and $\prec$ the lexicographic ...

**2**

votes

**2**answers

165 views

### Degree Sequence Problem on $k$-Partite Graphs

The general Degree Sequence Problem asks for a simple undirected graph (that is a graph without self-loops and with no more than one edge between any pair of nodes) for which it holds that the degrees ...

**0**

votes

**0**answers

30 views

### “Mutant knots” generalizable to “mutant tangled graphs”?

Just in case: Take a link L (drawn into the plane with over- and undercrossings),
draw a closed loop C on it which cuts L in four points, rotate the inside of C
around 180° (align the cut points on ...

**5**

votes

**0**answers

84 views

### 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 ...

**3**

votes

**1**answer

358 views

### Is there a graph with 99 vertices in which every edge belong to a unique triangle and every nonedge to a unique quadrilateral?

99-Graph:Is there a graph with 99 vertices in which every edge(i.e. pair of joined vertices) belong to a unique triangle and every nonedge(pair of unjoined vertices) to a unique quadrilateral?

**2**

votes

**1**answer

57 views

### Graph construction to double coloring & Hadwiger number

For any graph $G$ let $\eta(G)$ be the Hadwiger number of $G$.
Is there for every graph $G$ a graph $2G$ such that
-- $\chi(2G) = 2\chi(G)$, and
-- $\eta(2G) = 2\eta(G)$?
For each one of the above ...

**0**

votes

**1**answer

72 views

### Additivity of genus of a graph [duplicate]

Let $G$ be a finite simple undirected graph. Suppose there exist subgraph $G_1,G_2,\dots,G_n$ of $G$, such that $E(G_i)\cap E(G_j) = \emptyset$ and $|V(G_i)\cap V(G_j)| \leq 2$, for $i\neq j$. Then, ...

**0**

votes

**0**answers

154 views

### Induced graphs of cayley graph

I have a Cayley graph $Cay(G,S)$, its group presentation $G=< S | R >$ and it is a metric graph by assigning a length equal to 1 to each edge. I also have an induced subgraph of that Cayley ...

**1**

vote

**0**answers

65 views

### Recovering Spherical Harmonics from Discrete Samples

Consider a collection of $N$ points on the 2-sphere chosen uniformly at random. Let's say that there's an edge between two such vertices if their geodesic distance is less than $r_N$. The resulting ...

**3**

votes

**2**answers

206 views

### Definition of the Moebius Ladder Graph

I found two different definitions of the Moebius Ladder Graph, whose essential difference is, whether the smallest one shall be $K_4$ or $K_{3,3}$.
according to Wikipedia ...

**0**

votes

**0**answers

52 views

### Preferential Attachment and salton similarity in directed networks

Preferential Attachment similarity between two nodes in an undirected graph is the degree of the first node multiplied by the degree of the second node. But what about directed graphs? Which degree ...

**1**

vote

**1**answer

85 views

### Is undirected short-simple-path-through-3-vertices decidable in polynomial time?

Consider the following language:
$L=\{\langle G=(V,E),s,v,t,l\rangle\;|\;s,v,t\in V, l\in \mathbb{N} \wedge $ There exists a simple path from $s$ to $t$, going through $v$ of length $\leq l\}$.
($G$ ...

**2**

votes

**2**answers

133 views

### Which graphs generate a matroidal independence complex?

The independence complex $I(G)$ of a graph $G=(V,E)$ has as point set the vertex set $V$ and as simplices the independent sets of $G$.
Now, if $G$ is a well-covered graph (where all maximal ...

**0**

votes

**0**answers

43 views

### Almost symmetric route in digraphs

Let $D=(V,E)$ be a digraph. A route of length $k$ in $D$ is a pair $L=(S,\sigma)$, where $S=(s_1,s_2,\dots,s_{k+1})$ is a sequence of $k+1$ elements of $V$, and ...

**1**

vote

**0**answers

56 views

### Disconnecting a three regular graph

Setting, a random three regular graph. If I start removing edges, what's the probability that I disconnect it? What I know is that if I disconnect a graph I get a subgraph, and that the expected ...