**0**

votes

**0**answers

12 views

### Automorphism of a restricted irregular graph class

Motivation:
This query is motivated by this question . It has relation to the complexity analysis of this post.
I have been informed Highly Irregular Graph has number of automorphism $\leq n ...

**26**

votes

**3**answers

978 views

### Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

I've known that one can arrange all the numbers from $1$ to $\color{red}{15}$ in a row such that the sum of every two adjacent numbers is a perfect square.
$$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$
...

**5**

votes

**1**answer

240 views

### Terminology in combinatorics

I met the following two combinatorial concepts during a study outside of combinatorics. I am wondering if there are common terminologies in combinatorics.
A finite graph $G$ has the following ...

**-4**

votes

**0**answers

38 views

### is graph coloring problem in general np-complete?(solvable) [on hold]

graph coloring problem
Hi,
i tried to find an algorithm for this problem and i want to make sure. i found it with this knowledge.
1.is graph coloring problem in general to find the ...

**0**

votes

**2**answers

41 views

### Connectivity of weighted graph and zero Laplacian eigenvalues

Given an undirected graph $G$, and let $V$ denote its set of vertices and $E$ its set of edges. Suppose that there are no edges connecting the same vertex, and no more than one edge connecting any ...

**2**

votes

**0**answers

156 views

### Partitioning graph for Graph Isomorphism

Motivation: I am studying graph isomorphism problem. I am trying to construct a partitioning method to reduce search cases .
Construction:
$G$ is a $r$ regular graph, $k$ connected( not a complete , ...

**10**

votes

**3**answers

812 views

### A question on representation of graphs

Take a complete graph $K_n$. You want to assign a vectors from $\Bbb F_2^d$ to every edge such that sum of vectors in every simple cycle does not sum to $0$ vector. The question is what is minimum $d$ ...

**1**

vote

**1**answer

349 views

### Possibility of Disconnected Subgraphs of a $k$ Connected $r$ regular Graph under a given condition

Context: Given a adjacency matrix A of a $r$-regular graph $G$ (not complete graph $K_{r+1}$) . $G$ is $k$ connected.
The matrix A can be divided into 4 sub-matrices based on adjacency of vertex $x ...

**1**

vote

**0**answers

28 views

### Modularity in a graph - definition of the random component

This question concerns the definition of modularity in a graph.
Consider a simple, undirected, unweighted graph $G$ with vertices in set $V$ and edges in set $E$ . Between any two vertices there is ...

**4**

votes

**2**answers

175 views

### A generously vertex transitive graph which is not Cayley?

A graph is vertex transitive if $x \mapsto y$ by an automorphism.
A graph is generously vertex transitive if $x \mapsto y \mapsto x$ by an automorphism.
Simple facts:
GVT $\rightarrow$ unimodular. ...

**5**

votes

**1**answer

452 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 ...

**6**

votes

**1**answer

56 views

### Analysis of the Laplacian of a random bipartite graph

My analysis of an engineering problem reduced to analysis of the Laplacian of a (random) bipartite graph. There are a few particular questions I am interested in, but not sure which direction to take ...

**0**

votes

**1**answer

201 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 ...

**0**

votes

**1**answer

103 views

### Decomposition of a regular graph and connected subgraphs

I have asked almost same question earlier. I have been told that my question was poorly written, so I am trying to write it more clearly in this post. Also, this time I would be a little different in ...

**0**

votes

**0**answers

43 views

### Crossing all boundaries on a map? [closed]

In a variation on the traveling salesman problem, is there an algorithm (an approximate heuristic is fine) that finds a short, if not the shortest, path that crosses all boundaries between each pair ...

**0**

votes

**0**answers

64 views

### A constrained minimum edge coloring

Is minimum number of colors needed to color edges of complete graph $K_n$ so that every even simple cycle contains at least one color assigned to odd number of edges at most $\beta n$ where ...

**0**

votes

**0**answers

36 views

### Counting the k-factors of the complete graph on n vertices [closed]

I was originally trying to solve the following problem:
10 people are in a room, and you give them a task. Their task is for each person to shake hands with exactly 3 other people in the room. How ...

**2**

votes

**2**answers

52 views

### Minimal edge color on constraints

Is minimum colors needed to assign colors to edges of complete graph $K_n$ so that every even simple cycle contains an odd number ($>1$) of colors much larger than $(\log n)^\beta$ or ...

**2**

votes

**0**answers

50 views

### Even cycle constrained edge coloring

Is minimum colors needed to assign colors to edges of complete graph $K_n$ so that every $2t$ simple cycle where $t\in\Big\{1,\dots,2\Big\lfloor\frac{n}2\Big\rfloor\Big\}$ contains atleast $t+1$ ...

**4**

votes

**0**answers

78 views

### Does squaring a directed random graph more than double its out-degree?

As far as I know, it is an unsolved question
whether or not this is true:
If $G$ is a directed an oriented graph, $G^2$ always has some node whose outdegree is at least
double that of its ...

**1**

vote

**0**answers

221 views

### Incidence geometry and matrices

Supposing I have a $0/1$ or $\pm1$ matrix $A$ of size $m\times n$, is there a minimum $d$ (that works for every $m\times n$ $A$) such that there exists $m$ lines $r_1,\dots,r_m$, $n$ lines ...

**1**

vote

**0**answers

75 views

### Efficiently counting all paths of length n in a graph with vertex visitation contraints [closed]

I have a graph G with two classes of vertices. The first class represents no resource limitation entities and can be visited an unlimited number of times in any path traversal. The second class of ...

**0**

votes

**1**answer

182 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 ...

**8**

votes

**2**answers

225 views

### Dividing the edges and diagonals of a polygon among disjoint sub-polygons

Let $P$ be a convex $n$-gon ($n$ is odd and $n \geq 5$).
Determine the smallest $m$ such that all edges and diagonals of $P$ can be covered by the edges
of $m$ convex sub-polygons of $P$ which ...

**2**

votes

**2**answers

128 views

### “Nice” and “nasty” partitions in graphs

Let $G=(V,E)$ be a simple, undirected graph, that is $V$ is a set and $E \subseteq [V]^2 = \{\{v,w\}: v,w \in V \land v\neq w\}$.
For $v\in V$ and $S\subseteq V$ we set $$N(v,S) = \{w\in S: \{v,w\} ...

**5**

votes

**1**answer

145 views

### Modification of matching

Suppose i have an $n \times n$ random bipartite graph and suppose that i repeat the following process $n$ times. At the start (stage 1) each edge is selected independently with probability $p(n)$, and ...

**1**

vote

**0**answers

28 views

### One sided Satisfactory Partition problem

The Satisfaction Partition problem is to decide if a graph has a vertex partition (U,V) into non empty parts where each vertex has as many neighbours in its part as in the other part. This problem is ...

**1**

vote

**2**answers

107 views

### Counting the orderings of outward-directed trees where the degree of each vertex is $2$

Let $T$ be a connected directed tree with the following properties:
The degree of each vertex of $T$ is at most $2$ (I am sure there is a name for such a graph but I do not know it).
$T$ has a ...

**3**

votes

**1**answer

154 views

### Coloring algorithm maximising color difference between neighbors

Consider a graph and a set of ordered colors ${\cal C} = \{1,2,\cdots,C\}$. I want to color each node $i$ with a color $c_i\in{\cal C}$ so as to maximize the minimum color difference between two ...

**5**

votes

**0**answers

169 views

### Unique Nash equilibrium games

Multicast network design game is a special case of a general network design game (http://www.cs.cornell.edu/home/kleinber/focs04-game.pdf) in which there is a target vertex $t$ and $n$ rational ...

**1**

vote

**0**answers

33 views

### Weak law for component count of Erdos-Renyi random graphs

Penrose and Yukich derive a weak law for functionals of binomial point processes, which implies a law of large numbers for the component count of random geometric graphs. Do similar results exist for ...

**1**

vote

**0**answers

94 views

### From Planar Graphs To Tangent Circles

I have a conjecture:
"For each planar graph with vertices $V_1, V_2,\ldots, V_n$ there exist disjoint circles $w_1,w_2,\ldots,w_n$ in the plane, such that for every $i,j$, $w_i$ is tangent to $w_j$ ...

**6**

votes

**1**answer

345 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 ≥ ...

**2**

votes

**1**answer

60 views

### Images of interval edge coloring

I found out the definition of interval edge colorings, concept put by Kamalian in various papers but could not find a graph depicting an example. Where can I find pictures of explicit examples of ...

**8**

votes

**1**answer

197 views

### Smallest strongly regular graph whose automorphism group is not vertex transitive?

I'm looking for a small strongly regular graph whose automorphism group is not vertex-transitive.
This answer to a different question shows that the Chang graphs on 28 vertices are such graphs. Is ...

**12**

votes

**1**answer

434 views

### Drawings of complete graphs with $Z(n)$ crossings

Hill conjectured that the minimum number of crossings in a drawing of the complete graph $K_n$ in the plane is exactly
$$Z(n) = \frac{1}{4} \bigg\lfloor\frac{n}{2}\bigg\rfloor ...

**12**

votes

**1**answer

149 views

### Reconstruction Conjecture: are almost all digraphs reconstructible?

The Reconstruction Conjecture for simple graphs remains unresolved. Most attempts I've seen at resolving the conjecture aim at proving it to be true (or partially true). I don't believe there is a ...

**0**

votes

**1**answer

97 views

### Petersen 2-factor decomposition theorem for directed graphs

Petersen proved that every 2k-regular graph can be decomposed into k disjoint 2-factors. I would like to know that is it true that if G is a directed regular graph (d_out(v)=d_in(v)=k), then can G be ...

**1**

vote

**0**answers

109 views

### Cayley graphs with special subgraphs and some related problems

I asked some questions about finite Cayley graphs with special type of subgraphs which has been answered by Dear Prof. Godsil. It can be seen in the MO page with address:
Cayley graphs and its ...

**0**

votes

**1**answer

103 views

### Two graph structures on $\text{Hom}(G,H)$

By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq [V]^2 := \{\{a,b\}: a\neq b \in V\}$. A graph homomorphism between graphs $G, H$ is a map $f:V(G)\to V(H)$ such that $\{v, ...

**1**

vote

**0**answers

55 views

### Interpreting (Fiedler) spectral bisectioning

I would appreciate help on how to interpret the results of spectral bisectioning of a graph.
Given a $G=(V,E)$ with size $N$ represented by $Q$ its Laplacian matrix where the eigenvalues are ordered ...

**0**

votes

**2**answers

113 views

### Generate all non-isomorphic partitions $\pi = \{ \{1, …, n-1\}, \{n\} \}$ for all graphs of order $n$

Let $G$ be any connected, undirected, and unweighted graph of order $n$.
Let $\pi = \{ \{ 1, ..., n-1 \}, \{ n \} \}$ be partitioning of $G$ such that always $n-1$ vertices are in the first cluster ...

**12**

votes

**1**answer

865 views

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

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

**19**

votes

**5**answers

2k views

### The Matrix-Tree Theorem without the matrix

I'm teaching an introductory graph theory course in the Fall, which I'm excited about because it gives me the chance to improve my understanding of graphs (my work is in topology). A highlight for me ...

**5**

votes

**1**answer

83 views

### Resource Constrained Routing with Refueling

What are good algorithms (resp. models) for calculating optimal or near optimal routes while taking into account fuel consumption, options for refueling and, limited tank capacity?
Especially modeling ...

**20**

votes

**1**answer

248 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

**0**answers

45 views

### Existence of certain graph gadget related to coloring odd hole free graph

Wondering about the existence of graph gadget related to coloring
(or 3-coloring) odd hole free graphs.
Let $G$ be simple $k$-chromatic connected graph with two
vertices $u,v$.
Is it possible $G$ to ...

**1**

vote

**0**answers

139 views

### Complexity of reordering a matrix which consists independent sub matrices

Introduction:
Given a matrix A of a $k$ regular graph G. The matrix A can be divided into 4 sub matrices based on adjacency of vertex $x \in G$.
$A_x$ is the symmetric matrix of the graph $(G-x)$, ...

**3**

votes

**1**answer

158 views

### “Canonical” graph structure on $\text{Hom}(G, H)$

By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq [V]^2 := \{\{a,b\}: a\neq b \in V\}$. A graph homomorphism between graphs $G, H$ is a map $f:V(G)\to V(H)$ such that $\{v, ...

**8**

votes

**2**answers

344 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) = ...