**0**

votes

**0**answers

55 views

### A certain Acyclic Partition of a digraph

Has the following object been defined in the literature? What is it called? And what literature studies it? Are there other characterizations of this? What properties are known?
Let $G$ be a directed ...

**1**

vote

**0**answers

40 views

### Programmatically recognizing symmetries of a polyhedron

I asked this question on MSE a month ago, but nobody was able to answer it, so I guess the question is more difficult than I initially thought: ...

**2**

votes

**1**answer

60 views

### Minimum number of transitive paths in tournament

Let $T$ be a tournament with $n$ vertices (i.e., between every pair of vertices there exists an edge in exactly one direction.) For any $k$, the vertices $A_1,A_2,...,A_k$ form a transitive path if ...

**0**

votes

**1**answer

131 views

### Efficient isomorphic subgraph matching with similarity scores

I'm a computer vision PhD student, and I'm looking for an efficient approximation to the following problem, which could end up helping in image to image matching. Failing that, pointers to relevant ...

**1**

vote

**0**answers

68 views

### Kempe chain color swaps in a partially colored map

Crossposted from math.stackexchange.com:
http://math.stackexchange.com/questions/904932/kempe-chain-color-swaps-in-a-partially-colored-map
Question: In this partially Tait's colored map, using ...

**8**

votes

**1**answer

335 views

### What is the correct statement of this Erdös-Gallai-Tuza problem generalizing Turan's triangle theorem?

In Zsolt Tuza's Unsolved combinatorial problems I, Problem 46 is the following conjecture:
Let $G$ be a graph on $n$ vertices. Let $\alpha_1$ be the maximum number of edges of $G$ such that every ...

**-1**

votes

**0**answers

55 views

### How to find number of nonisomorphic simple graphs from a number of edges and a number of vertices? [on hold]

How many nonisomorphic simple graphs are there with __ vertices and _ edges?
(Fill in the blanks with numbers, say 6 and 4)
I've tried to use incident matrices to no avail.

**2**

votes

**1**answer

60 views

### When does a hypergraph represent maximal independent sets?

Let $G = (V,E)$ be a simple graph. Then, we can view the set of maximal independent sets (or the set of maximal cliques) as a hypergraph $H = (V, E')$. This is quite a useful device when connecting ...

**4**

votes

**1**answer

209 views

### How to find or constrain “particularly good” (two-sided) spectral expanders?

I'm new to graph theory, but a response to a question I asked a while ago introduced me to the concept of expander graphs.
A k-regular graph (henceforth "graph") on n nodes has eigenvalues k = λ1 ≥ ...

**7**

votes

**1**answer

171 views

### A conjecture about strongly regular graphs

Let $G \neq K_{v}$ be a $(v,k,\lambda,\mu)$ strongly regular graph. After perusing through Brouwer's tables of parameters I have formed the conjecture $$\lambda-\mu \leq \frac{k}{2}.$$
So far I have ...

**25**

votes

**3**answers

575 views

### Removal of non-isomorphic edges results in the same graph

There exists a (simple unlabeled) graph on 6 nodes with a pair of non-isomorphic edges (i.e., there is no graph automorphism that sends one edge into the other) such that removal of either of them ...

**1**

vote

**1**answer

95 views

### edge graph reconstruction conjecture : set vs multi set

Why is the edge reconstruction conjecture stated with the deck defined as the multi set of graphs formed by deleting one edge? Can someone give an example of two graphs such that the edge deleted ...

**0**

votes

**2**answers

110 views

### Terminology for beads/necklace/bracelet problem [closed]

I'm new to mathoverflow but hopefully anyone here can point me in the right direction.
The problem is as follows, imagine you have 4 beads, lets give them numbers 1,2,3,4. Now I want the unique ...

**2**

votes

**1**answer

77 views

### What is the minimal girth of a cayley graph for Alt(n) in which the girth relator is not a proper power?

First, a definition: a girth relator for a Cayley graph is a word that you get by reading the edge labels along a shortest loop. The girth relator is not usually unique, though it is often unique up ...

**3**

votes

**0**answers

81 views

### Full-rank factorization of the graph Laplacian

Is there a combinatorially meaningful full-rank factorization of the Laplacian matrix of a graph?
The usual factorization $L=BB^{T}$, where $B$ is an oriented incidence matrix, is full-rank if and ...

**10**

votes

**1**answer

583 views

### A generalization of the triangle counting problem for simple weighted graphs

One nice identity is $$tr(A^3)/6$$ which counts the number of triangles of a graph represented with adjacency matrix $A.$ It also implies that triangle counting can be performed in subcubic time.
...

**3**

votes

**1**answer

130 views

### Contracting a planar graph to a (multiply-edged)-tree

Given a planar graph (no loops, no multiple edge), is it always possible to perform edge contractions* in order to obtain a graph $T$ which has no loops, but if one forgets the multiplicity of its ...

**6**

votes

**1**answer

155 views

### Duration and critical groups order in sandpile models and chip firing games

The famous chip firing game (which is closely related to sandpile models) goes like this:
Place chips at the vertices of a graph. REPEATEDLY: If a vertex $v$ of
degree $d_{v}$ has at least ...

**8**

votes

**1**answer

155 views

### Looking for history on a theorem of clique intersections

I have a short paper I'm working on where I prove:
Theorem: Every graph on (2t-1) vertices with no (t+1)-clique has a vertex that is contained in every t-clique.
By "t-clique", I mean a complete ...

**5**

votes

**2**answers

227 views

### Matching number and chromatic number

If $G$ is a (finite) graph, denote with $\mu(G)$ the size of any maximum matching in $G$ (this number is also called the "matching number" of $G$).
For odd integers $n$ we have $n=\chi(K_n) = ...

**5**

votes

**1**answer

290 views

### How does the distribution of Erdős number evolve over time ? How to build a model to fit the real data ?

Let $E(n,t)$ be the number of mathematicians with finite positive Erdős number $n$ at time $t$. As old mathematicians leave, and new mathematicians come, how does $E(n,t)$ change over time ?
We can ...

**0**

votes

**1**answer

40 views

### average number of cycles and closed walks length k in incomplete directed graph

I asked this question before, but formulation was poor. I've deleted previous question and reformulate it again.
Let graph $G=(N,p)$ is finite simple incomplete directed graph of size $N$ (multiple ...

**2**

votes

**1**answer

146 views

### Cospectrality and dimension of graphs

Firstly, I apologize if the question is long. I appreciate any helpful answers and ideas.
In the following all graphs are simple and connected.
Let $G$ be graph with vertex set ...

**2**

votes

**0**answers

46 views

### Looking for similar centrality measurement on graph

I'm working on a graph problem somehow related to centrality measurement. Given an undirected, unweighted tree $T$ and a vertex $v$, let $D_i(v)$ be the set of vertices in $T$ that are i hops from ...

**14**

votes

**0**answers

567 views

### A Conjecture About Directed Graphs that are the Union of Two Trees

Let D=(V,E) be a directed graph that is the union of two edge-disjoint directed
spanning trees. Suppose that
There no subset X of vertices so that
there is precisely one directed edge
from X ...

**5**

votes

**0**answers

70 views

### Mapping graphs to ordinals

Robertson-Seymour theorem implies that graph minor relation is a well-quasi-ordering, which means (among other things) that this relation can be extended to a well-order, and other result says that ...

**0**

votes

**1**answer

78 views

### Finding node-disjoint routes with mutually exclusive nodes in graphs

I have the following problem. I would like to know if it reduces to some standard problem in Graph theory. Particularly, I would like to know whether it is NP-hard, if yes, how to prove its ...

**5**

votes

**1**answer

343 views

### Flow of an integer

I've stumbled across this family of flow networks, and posted the sequence of maximal flows to OEIS: A238729. I can't find any reference to it either. Has anyone seen it?
Here is the description:
...

**11**

votes

**0**answers

695 views

### Hamiltonian cycles and fundamental groups

I'm interested in the interplay between the Hamiltonian cycles of graphs and the compact surfaces they embed in. I was doing some reading on the Lovász conjecture for Cayley graphs, I started noticing ...

**17**

votes

**1**answer

521 views

### Monomer-Dimer tatami tilings need better relationships with other math. Summary of results.

A monomer-dimer tiling of a rectangular grid with $r$ rows and $c$ columns satisfies the \emph{tatami} condition if no four tiles meet at any point. (or you can think of it as the removal of a ...

**5**

votes

**2**answers

176 views

### Cubic graphs whose 2-factors all have the same cycle type

Let $G$ be a bridgeless cubic graph. I am interested in such graphs where all 2-factors are isomorphic (as graphs), i.e. have the same partition as cycle type. We'll say that this partition is ...

**2**

votes

**0**answers

47 views

### When polynomial GI implies polynomial (edge) colored GI?

(edge) colored graph isomorphism is GI which
preserves the colors (of edges if it is edge colored).
There are several reductions using transformations/gadgets
of (edge) colored GI to GI. For edge ...

**2**

votes

**0**answers

55 views

### Properties of a smallest tournament with domination number $k$

For some tournament $T$, let $\gamma(T)$ denote the cardinality of a smallest dominating set of $T$.
Denote by $f(k)$ the minimum number of vertices of a tournament $T$ having $\gamma(T) = k$.
From ...

**2**

votes

**0**answers

110 views

### NP-hard proof of optimization version of exact cover [closed]

Exact cover is NPC.
http://en.wikipedia.org/wiki/Exact_cover#Equivalent_problems
Given a collection $\mathcal{S}$ of subsets of a set $X$, an exact cover is a >>subcollection $\mathcal{S}^*$ ...

**4**

votes

**1**answer

225 views

### If a graph invariant is NP-Hard, is its “deck ratio” NP-Hard as well?

This question is inspired by the Graph Reconstruction Conjecture. Suppose that $\psi$ is some graph invariant and that it is NP-Hard. There is a plethora of examples, of course. Now define ...

**5**

votes

**1**answer

297 views

### Bipartite Graphs arising from two k-partitions of a given Graph

Let $G$ be an $n$-chromatic connected graph. Let $(V_1, V_2, \cdots, V_n)$ and $(U_1, U_2, \cdots, U_n)$ be two partitions of $V(G)$ corresponding to proper n-colorings of $G$.
Consider the bipartite ...

**0**

votes

**1**answer

61 views

### N random walkers that hit node v in a graph

Consider a finite, undirected graph G, with uniform edge weights. Assume that there are n number of random walkers that will start at different nodes (lets say n=3, hence the random walkers will start ...

**2**

votes

**1**answer

172 views

### Coloring vertices in a cubic lattice graph and counting edges between vertices of identical and vertices of distinct coloration

Take an $A \times B \times C$ cubic lattice graph $G$, and paint $k_1$ vertices with color $c_1$ & $k_2$ vertices with color $c_2$, where $(k_1 + k_2)$ is equal to the total vertex count. Let ...

**0**

votes

**1**answer

111 views

### Reverse optimization of a minimum cost flow network

Given an undirected graph $(V,E)$, with $W$ as the weight of each edge, and a convex cost function $F(X)$, such as $|X|^k$ ($k>1$).
The cost to send $x$ unit of flow through edge $e_i$ is defined ...

**1**

vote

**1**answer

61 views

### Two definitions of genus for circle graphs

In the (very nice) article of Goldstein and Turner untitled Applications of Topological Graph Theory to Group Theory, the following definitions can be found:
Definitions: A circle graph is a pair ...

**6**

votes

**1**answer

138 views

### Smallest Connected Graph for Given Degree Sequence

For a given integer sequence $(d_1, d_2,...,d_n)$, a natural question is if such a sequence is graphical, i.e. is a degree sequence of some graph. According to Erdős–Gallai theorem, A sequence of ...

**6**

votes

**1**answer

54 views

### Is a connected graph uniquely determined by its weighted 2-step graph?

This is an extension of a previous question: Is a graph uniquely determined by its weighted 2-step graph?. In that question I asked about arbitrary graphs; in this question I restrict to connected ...

**7**

votes

**2**answers

247 views

### What is the number of noncrossing acyclic digraphs?

A noncrossing graph on $n$ vertices is a graph drawn on $n$ points numbered from $1$ to $n$ in counter-clockwise order on a circle such that the edges lie entirely within the circle and do not cross ...

**3**

votes

**1**answer

209 views

### How hard is a variant of graph automorphism problem?

I'm interested in a variant of graph automorphism problem (which is prime candidate for $NP$-Intermediate problem).
Restricted GA
Input: Given an undirected graph $G(E, V)$, and $\epsilon |V|/2$ ...

**0**

votes

**1**answer

41 views

### Resource on characterizations or properties of traceable graphs

I am looking for some resources that provide information on traceable graphs(paths containing a hamiltonian path). I have found a lot of information on hamiltonian graphs, but none on traceable ...

**2**

votes

**0**answers

158 views

### Computing the chromatic polynomial of graph modulo $x-3$

The chromatic polynomial of graph $P(G,x)$ is univariate
polynomial which counts the number of colorings of $G$
with $x$ colors for natural $x$.
Graph is not $k$ colorable iff $P(G,k)=0$.
The ...

**0**

votes

**0**answers

69 views

### asymptotic notation with graph colouring

This is my first ever post so I hope this is an appropriate question.
Basically I am looking at the paper here: http://homepages.math.uic.edu/~mubayi/papers/biclique.pdf
Namely theorem 5.
Now, feel ...

**30**

votes

**5**answers

2k views

### Does every bipartite graph with 512 edges have an induced subgraph with 256 edges?

Suppose we have a (simple) bipartite graph with $2^k$ edges. Is it true that there is a subset of the vertices such that their induced subgraph has exactly $2^{k-1}$ edges?
I know that the answer is ...

**11**

votes

**0**answers

370 views

### Is this graph polynomial known? Can it be efficiently computed?

I am a physicist, so apologies in advance for any confusing notation or terminology; I'll happily clarify. To provide a minimal amount of context, the following graph polynomial came up in my research ...

**6**

votes

**0**answers

66 views

### Set system with prescribed intersection sizes

Questions: What is the asymptotic maximal size of a $4$-uniform (every set has 4 elements) set system $\mathcal{A}$ of subsets of $[n]$ such that, no two sets have size of their intersection $2$?
In ...