# Tagged Questions

**2**

votes

**0**answers

56 views

### Problems Solvable/Decidable by Counting Shortest Paths in Graphs

This questions is based on a dispute, whether it would be possible to calculate 'nice' routes in Manhattan, if the road network is assumed to be a rectangular grid and, that 'nice' means that there is ...

**5**

votes

**1**answer

300 views

### Are there number-theoretic graphs that are far from being isomorphic

I say that two graphs $G_1=(V_1,E_1)$, $G_2 = (V_2,E_2)$
with the same number of vertices, edges,
are $\epsilon$-far from being isomorphic, if for any bijection between $V_1$ and $V_2$, the fraction ...

**0**

votes

**1**answer

87 views

### Comparing ideals in posets

Consider a partially ordered set $P$, and two upper sets $U_1$, $U_2$ in this poset.
What are some natural ways to measure how equal these two upper sets are?
This question arise naturally in the ...

**5**

votes

**0**answers

157 views

### (Connected) Cayley graphs of PSL(2,q) from (2,3,n)-triples

Let $G = PSL(2,q)$. I'm interested in the Cayley graphs of $G$ generated by triples $(A,BAB^{-1},B^{-1}AB)$, where $A, B \in G$ are elements of order $2, 3$ respectively: such a triple generates all ...

**1**

vote

**0**answers

73 views

### Disks in Flat Embeddings of Graphs in $\mathbb{R}^3$

Robertson, Seymour and Thomas proved that any linkless graph $G$ has a flat embedding in $\mathbb{R}^3$ (see for example A survey of linkless embeddings). An embedding of $G$ is flat if for any cycle ...

**2**

votes

**1**answer

62 views

### Hamiltonicity of random graphs with high girth

We say that $G\sim G_{n,f}$ (for $f=f(n)$) if $G$ is chosen uniformly at random from all graphs on $n$ vertices with girth $g(G)\ge f(n)$. Is there any threshold function $F(n)$ such that when $f\ll ...

**1**

vote

**0**answers

61 views

### Possible ways to create a graph representation from a distance matrix (through approximation)

Forgive me, Im not math professional, but a computer scientist at the beginning of my base research from my thesis, so bare with me if I miss something blatantly obvious.
I have a Euclidean distance ...

**3**

votes

**0**answers

61 views

### Different definitions of linkless graphs

Robertson, Seymour and Thomas defined linkless embeddings of graphs as follows:
An embedding of $G$ is linkless if every pair of disjoint circuits of $G$ have zero linking number (see here). However ...

**1**

vote

**1**answer

93 views

### Difference between straight and piecewise linear and continuous embeddings of graphs / complexes in d-dimensional space?

Here is my main question: what is the difference between "straight" and "piecewise linear" and "continuous" embeddings of graphs/complexes in d-dimensional space?
Moreover I would like to know if any ...

**2**

votes

**2**answers

350 views

### Categories with binary relations as objects

For the category of functions, pairs of functions making commutative diagrams are the canonical morphisms $\alpha:f\rightarrow g$. For binary relations there is an alternative, to consider the ...

**2**

votes

**1**answer

66 views

### Coloring graph such that the coloring classes are not maximal independent sets

Let $G$ be a (finite or infinite) simple graph. We let $\mathrm{Ind}(G)$ be the collection of independent sets. For any cardinal $\kappa$ and coloring map $\chi: G\to \kappa$ we have ...

**2**

votes

**0**answers

78 views

### second smallest eigenvalue Laplacian - submodular set function

Let $G$ be a connected unweighted undirected graph. In addition, let $\lambda_2(L(G))$ be the second smallest eigenvalue of the Laplacian matrix of graph $G$. Is $\lambda_2(L(G))$ a submodular set ...

**1**

vote

**2**answers

168 views

### Cubic Cayley (undirected) graphs

The generators of a cubic Cayley graph always include at least one involution, since 3 is odd. There are then two possibilties for the other two generators: either (A) they are also (distinct) ...

**0**

votes

**0**answers

55 views

### Programmatically recognizing symmetries of a polyhedron [on hold]

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: ...

**4**

votes

**2**answers

141 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 ...

**1**

vote

**0**answers

102 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 ...

**0**

votes

**0**answers

67 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 ...

**2**

votes

**1**answer

77 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 ...

**7**

votes

**1**answer

196 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 ...

**0**

votes

**2**answers

139 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

91 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 ...

**4**

votes

**0**answers

94 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 ...

**3**

votes

**1**answer

137 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 ...

**8**

votes

**1**answer

175 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

**1**answer

176 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 ...

**5**

votes

**2**answers

247 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) = ...

**0**

votes

**1**answer

53 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 ...

**5**

votes

**0**answers

81 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 ...

**2**

votes

**0**answers

49 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

48 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 ...

**2**

votes

**0**answers

64 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 ...

**1**

vote

**1**answer

66 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

**2**answers

209 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 ...

**6**

votes

**1**answer

159 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 ...

**2**

votes

**0**answers

126 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 $\mathcal{S}^*$ ...

**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 ...

**0**

votes

**0**answers

73 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 ...

**5**

votes

**1**answer

57 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 ...

**25**

votes

**2**answers

791 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 ...

**7**

votes

**0**answers

75 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 ...

**1**

vote

**1**answer

102 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 ...

**7**

votes

**2**answers

286 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 ...

**5**

votes

**1**answer

130 views

### Graphs where each edge belongs to the same number of 1-factors

Let $G$ be a simple connected graph that has at least one 1-factor. We'll define:
$G$ has property A iff it is edge-transitive.
$G$ has property B iff each edge belongs to the same number of ...

**2**

votes

**0**answers

166 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 ...

**1**

vote

**1**answer

129 views

### Vertex transitive and edge transitive and line graph

How can we find the proof of the following statement:
An undirected graph is edge transitive if and only if its line graph is vertex transitive.

**24**

votes

**3**answers

599 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 ...

**10**

votes

**2**answers

488 views

### Is every knot unavoidable in the embeddings of some graph?

Is it the case that, for any given knot $K$,
there exists some graph $G$ whose every embedding into $\mathbb{R}^3$
(or into $\mathbb{S}^3$)
contains a cycle that realizes $K$?
I know the ...

**2**

votes

**1**answer

217 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

**0**answers

43 views

### Complexity of graph isomorphism in $(P_4 \cup K_1,\overline{3K_2})$-free graphs

Related to this question where isomorphism preserving
transformation maps triangle-free graphs to $(P_4 \cup K_1,\overline{3K_2})$-free graphs.
What is the complexity of graph isomorphism in $(P_4 ...

**3**

votes

**1**answer

108 views

### Graph transformation related to graph isomorphism

Basically got graph transformation related to graph
isomorphism.
Define $G \to G'$. $V(G')=V(G) \cup E(G)=\{v_1\ldots v_n\} \cup \{e_1\ldots e_m\}$. Call $v_i$ vertices $v'$ and $e_i$ vertices $e'$.
...