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, ...

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0
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0answers
60 views

Constructing a digraph from its spectrum

This is related to the following question from cs theory stack exchange: http://cstheory.stackexchange.com/questions/3742/reverse-graph-spectra-problem So it seems as if given a sequence of real ...
6
votes
2answers
257 views

What is the maximum of the ratio $\vartheta(G)/\alpha(G)$?

A maximum independent set is a largest independent set for a given graph $G$ and its size is denoted $\alpha(G)$. And the Lovász number of $G$ is denoted $\vartheta(G)$. $\vartheta(G)\geq \alpha(G)$ ...
2
votes
2answers
125 views

Name for Kneser/Johnson-like graphs?

I wonder if the following simple generalization of Johnson and Kneser graphs has a name? Let the vertex set of the graph $G(n,k,t)$ be the set of $k$-element subsets of an $n$-set, with two $k$-sets ...
4
votes
3answers
166 views

Crystal structure, lattice, Graph and coloring

I am working across mathematics, physics and engineering. And I am looking for whether there exists already formally established knowledge in the field. Given a periodic graph (actually a physical ...
2
votes
3answers
381 views

Counting graphs up to isomorphism

I apologize if the questions are too elementary for the forum, but I am not an expert in this fields. How many (rooted or unrooted) binary trees with $n$ vertices are there up to isomorphism? How ...
0
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1answer
68 views

On homomorphisms between vertex transitive graphs

In general there is no relation between automorphism groups of subgraphs and the main graph. However, this question is about vertex transitive graphs. Given vertex transitive $G$ and $H$ such that ...
-1
votes
1answer
198 views

an interesting conjecture about even cycle

Let G be a simple graph which is a $2n$-cycle equipped with $n$ chords such that $G$ is $3$-regular,in other words,the set of the $n$ chords is a perfect matching of $G$(that is,every vertex of $G$ is ...
2
votes
1answer
117 views

Factors of Kneser graph

With respect to the Strong product, is the Kneser graph prime and if not how does one find a prime decomposition? Are there any references or algorithms?
1
vote
2answers
214 views

a conjecture about Hamiltonian graph

Suppose $G$ is a simple graph and $V(G)=V(C)\bigcup \{u_1,...,u_n\}$,where $C$ is a $2n$-cycle in $G$ and $V(C)=\{a_1,...,a_n,b_1,...,b_n\}$ such that $(1)V(C)\bigcap \{u_1,...,u_n\}=\varnothing$; ...
3
votes
1answer
391 views

Is there a graph-theoretical proof of Tutte's theorem on matroids?

First of all, I'm not a mathematician and I hope this question isn't too elementary, but I got no answers on math.SE, and since this is a reference request on a relatively advanced theorem, I thought ...
2
votes
0answers
106 views

The Turán problem for graphs with given chromatic number

The ordinary Turán problem for graphs asks, "Given a graph $H$, if $G$ is an $H$-free graph on $n$ vertices, what is the largest number of edges that $G$ can have?" As is well known, if $\chi(H) = r ...
2
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0answers
183 views

On the existence of Graph Monomorphism

A graph monomorphism is an injective graph homomorphism. Determining existence of Graph monomorphism between graph pairs is computationally hard. Assume we talk only about classes of undirected ...
5
votes
3answers
312 views

New trends in Applied Graph Theory [closed]

What are current trends in Applied Graph Theory? I am interested mainly in non-algorithmical problems. Maybe even in applications of graphs to other mathematical disciplines. For example, abstract ...
2
votes
1answer
155 views

positive semidefinite matrix condition

There is a great work of Alizadeh that in section 4 speaks about Minimizing sum of the first few(k-largest) eigenvalues of a symmetric matrix. Instead of a symmetric model we use the weighted ...
5
votes
1answer
415 views

No big clique minor but a big grid minor

I was wondering if the following result is known (or if there's a nice short proof without treewidth/brenchwidth related theorems): as the title says, suppose you have a graph without a big clique ...
9
votes
2answers
216 views

the length of cycles in a $2$-connected simple gragh

Let $G=(V,E)$ be a simple $2$-connected graph and $C$ is a cycle in $G$ satisfies: For any vertex $v$ of $C$,there exists at least one vertex $u\in V(G)\backslash V(C)$ adjacent with $v$. Is it true ...
5
votes
2answers
470 views

A conjecture about odd path and odd cycle

Let $k$ be a positive integer and $G=(V,E)$ be a $2$-connected simple graph.Suppose $v\in V(G)$ satisfy: $(1)$there exists at least one vertex $u\in V(G)\backslash\{v\}$ such that $u$ is not adjacent ...
3
votes
1answer
127 views

Triangulations of a disk, flip distance and hamiltonian circuit

Let us consider a disk with one labelled point on the boundary and $n$ labelled points in the interior. Let T be a triangulation of the whole disk with vertices on the labelled points such that T ...
0
votes
0answers
125 views

Monotone graph parameters under vertex deletion

Let $f(G)$ denote any parameter of a graph $G$ for which $f(G) \geq f(G - \{v \})$, where $v$ is any vertex in $G$. We could describe such parameters as being monotone under vertex deletion. Does ...
5
votes
0answers
162 views

A linear optimization problem on a graph

Let $G=(V,E)$ be a finite graph and let $f$ be any positive function defined on the vertices. Put weights on the vertices $v_{i}$, way $w_{i}$ so that $\sum_{i=1}^{n}w_{i}\leq 1$. Assume that every ...
7
votes
0answers
181 views

Is there a Rado category?

The Rado graph appears to have a nice universality property (it contains all finite and all countably infinite graphs as induced subgraphs) and homogeinety property (any isomorphism between ...
1
vote
0answers
66 views

How close to platonic can a non-platonic planar graph be?

Direct question: Is it possible to construct a finite, planar, $k$-regular graph in which all the faces except one have the same degree (are bounded by a cycle with the same number of edges), and the ...
2
votes
3answers
230 views

eigenvalue of Laplacian matrix

If we have a Laplacian matrix $\boldsymbol{A}$ such that \begin{align} &A_{ii} >0 \\ &A_{ii}=-\sum_{j\neq i}A_{ij} \end{align} with known eigenvalues $\lambda_i$. Define the matrix ...
0
votes
0answers
93 views

Adjacent matrix of undirected graph with a giant component

Assuming there is a undirected random graph $G=(V,E)$, $|V|=N$ and its adjacent matirx is $A$. What is the sufficient and necessary conditions of A for that there is a giant component of graph $G$? ...
9
votes
2answers
383 views

Is the Steiner ratio Gilbert–Pollak conjecture still open?

Gilbert-Pollak conjecture on the Steiner ratio: Consider a set $P$ of $n$ points on the euclidean plane. A shortest network interconnecting $P$ must be a tree, which is called a Steiner minimum ...
2
votes
1answer
182 views

When does graph Laplacian have eigenvalue -1?

Consider an undirected graph $G$ with (symmetric) adjacency matrix $A \in \{0,1\}^{n \times n}$ and degree sequence $d = (d_i)$ where $d_i = \sum_{j} A_{ij}$. Assume that every node has degree at ...
0
votes
1answer
117 views

3-complexes not embeddable in 3-space

My question is about embeddability of 3-dimensional complexes in R^3. Do we have something like Kuratowski's theorem for complexes in 3-space which specifies a set of minors for non-embeddability?
1
vote
0answers
64 views

maximum weight k-edge problem

Given positive integer $k$ and an undirected graph $(V,E)$, with nonnegative (non-uniform) weights on the nodes. Find $k$ edges whose spanning nodes have the maximum weight. Is this in P or NP? I ...
2
votes
0answers
87 views

Capacity of Cycle Graphs

Shannon capacity $\Theta(G)$ of pentagon is achieved at $2$-fold strong product of the pentagon. It is also known that the Lov\'asz theta $\vartheta(G)^m\neq\alpha(G^{\boxtimes m})$ for any finite ...
9
votes
0answers
160 views

How many n/2-cycles can a cubic graph have

Given a simple cubic graph with $n$ vertices (which implies that $n$ is even), what is a good upper bound on the number of cycles of length $n/2$ it can have? A random cubic graph has ...
7
votes
1answer
439 views

Separating pairs of points in R^n

Let $A$ be a set of $2k$ points in $\mathbb{R}^n$ such that no open set in $\mathbb{R}^n$ of diameter $2$ contains more than $k$ of these points. What is the largest possible distance $r_n>0$ one ...
2
votes
0answers
84 views

Groups of automorphisms of weighted graphs

Let $\Gamma=(V,E,\omega)$ be an (edge-)weighted graph without loops and multiple edges. Here $V$ is the set of vertices, $E$ is the set of edges and $\omega:E \to \mathbb{N}$. A permutation $\varphi$ ...
5
votes
0answers
451 views

Does this graph property have a name?

I'm interested in a family of properties of connected simple graphs that comes up in percolation theory. Let $G$ be a simple connected graph. Now consider the set of subgraphs of $G$ that I will call ...
4
votes
0answers
137 views

Graphs with many positive eigenvalues of their distance matrix

Let $G$ be a simple connected graph $D(G)$ its distance matrix and $n_{+}(G), n_{-}(G)$ the number of positive and negative eigenvalues of $D(G)$ respectively. We call a graph $G$ optimistic if ...
2
votes
1answer
69 views

Graphs with polynomial volume growth

Let $G = (V,E)$ a graph, equipped with graph distance (i.e. for $x,y \in V$, the distance $d(x,y)$ is the length of the minimum path connecting $x$ and $y$). For $x \in V$ and $r \in \mathbb N$, ...
7
votes
0answers
132 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 ...
11
votes
0answers
237 views

A Ramsey avoidance game

Consider the following game: Given $K_n$ the complete graph on $n$ vertices, two players take turns coloring its edges. Initially no edges are colored. At his turn a player can color a prevoiusly not ...
5
votes
5answers
254 views

Efficient Hamiltonian cycle algorithms for graph classes

Generally speaking finding a Hamiltonian cycle is NP-Hard and so tough. But if $G=L(H)$ is the line graph of $G$ then we can reduce the finding of a Hamiltonian cycle in $G$ to a Eurler your of $H$ ...
5
votes
1answer
144 views

NP-hardness of sparsest cut

Consider bipartitioning the vertices of a graph $(V,E)$ into $V = P \cup Q$ to minimize $$\frac{|E(P,Q)|}{|P| |Q|},$$ where $E(P,Q)$ denotes the set of edges in the cut. The usual citation for ...
1
vote
0answers
101 views

Is the automorphism group of a homogeneous (locally finite) tree unimodular?

I have seen somewhere (that I don't remember now) that the (full) automorphism group of a k-regular tree is unimodular. I assume a k-regular tree is the same thing as the homogeneous tree of degree k ...
3
votes
1answer
149 views

vertex independent set and the maximal clique

Let $N>2$ be a positive integer and $G$ be a simple graph satisfies: the maximal degree of $G$ is $N$ the clique number of $G$ is $N$. I want to ask if there exists a vertex independent set $I$ ...
2
votes
2answers
180 views

Partition of $\mathbb{F}_2^n$?

Consider an undirected graph $G$ with $n$ nodes denoted by $i$, $i \in [n] = \{1,2,...,n\}$. Denote the set of neighbours of node $i$ in the graph by $N(i)$. Given that there exists a set ...
1
vote
0answers
97 views

A traveling time problem

Given any undirected, connected and simple graph $G(V,E)$,each node of which is considered as a city. We call $j$ a neighbor of $i$ if $(i,j)\in E$. $N_i$ is the set of neighbors of $i$. $|V|=N$ ...
8
votes
1answer
264 views

normalized laplacian spectrum of trees

Is it known for which class of graphs the normalized laplacian has only simple eigenvalues (i.e., with multiplicity one)? In particular, are there trees (or perhaps a specific class of trees) whose ...
2
votes
0answers
51 views

Upper bound on size of obstruction set for wye-delta-wye reducible graphs

A graph is $Y \Delta Y$-reducible if it can be reduced to an empty graph by the following operations: $Y \leftrightarrow\Delta$ transforms; Replacing multiple edges with single edges (parallel ...
3
votes
0answers
108 views

Concentration inequality for function of independent Bernoulli r.v.'s (related to random graph)

Consider a random undirected graph on a set of $n$ nodes, say $\{1,2,\ldots,n\}$, such that the probability of edge between nodes $i$ and $j$ is $p_{ij}$ (we may assume $p_{ij}=o(1)$ for all $i,j$, ...
3
votes
1answer
130 views

the minimum possible value of the order of a graph G which is a finite union of N-order complete graphs

Let $N$ be a positive integer,$G$ be a simple graph and $H_1,H_2,\ldots,H_k$ be a family of subgraphs of $G$ which satisfy: every $H_i$ is a $N$-order complete graph; the union of $H_i$ is $G$; the ...
3
votes
2answers
244 views

chromatic number of a simple graph whose length of the longest odd cycle is 2k+1

Let $G$ be a simple graph and the length of the longest odd cycle of $G$ is $2k+1$,then I guess the chromatic number of $G$ is no more than $2k+2$,is it right?
3
votes
1answer
63 views

Commensurability of 2-colorings of finite 4-valent graphs

It is quite easy to show that given two finite 4-valent graphs $X,X'$ (I will take the convention that there is at most one edge between two vertices, but allow loops) there is a third such graph ...
5
votes
2answers
138 views

Embedding points in 2D based on distance estimates?

Suppose we have a collection of exactly $N$ points (say $N=1000$), with each point belonging to 2-dimensional Euclidean space $\mathbb{R}^2$, but we don't know the coordinates of the points. Suppose ...