**2**

votes

**1**answer

33 views

### Simple decomposition of $K_{2n}-I$ into hamiltonian cycles

http://mathworld.wolfram.com/HamiltonDecomposition.html
In the 1890s, Walecki showed that complete graphs K_n admit a Hamilton decomposition for odd n, and decompositions into Hamiltonian cycles ...

**4**

votes

**0**answers

49 views

### Edge-colorings of plane graphs: do you know references where the following questions are studied?

Let $G$ be a plane graph (or more generally, a graph embedded on a surface) with a proper edge-coloring of $G$ with $k$ colors $\{1,\ldots,k\}$. I am interested in studying the cyclic permutations of ...

**-1**

votes

**1**answer

63 views

### Graphs such that contracting an edge decreases the chromatic number [on hold]

Let $G = (V,E)$ be a finite, simple, undirected, connected graph, such that contracting an edge reduces the chromatic number. Does this imply that $G$ is complete?

**1**

vote

**1**answer

131 views

### Is there anything similar to the four color theorem for 3-dimensional objects?

From Wikipedia:
In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more ...

**5**

votes

**1**answer

107 views

### Closeness graph of a topological space

Let $(X,\tau)$ be a topological space. We say that $x, y \in X$ are close if for every neighborhood $U$ of $x$ and $V$ of $y$ we have $U\cap V \neq \emptyset$. Let $E$ be the set of $\{x,y\}$ where ...

**-2**

votes

**0**answers

54 views

### Maximally nonplanar graphs [on hold]

Is there any way to characterize maximally non-planar graphs?
For example, given a random collection of k-regular graphs, there is a clear distinction between the planar and non-planar graphs by ...

**7**

votes

**1**answer

146 views

### A matching that covers vertices with maximum degree

We have a graph G with maximum degree $\Delta$. The induced subgraph on vertices with degree equal to $\Delta$ is a bipartite graph (while the original graph is not).
Prove that G has a matching that ...

**-2**

votes

**1**answer

69 views

### Shortest path problem [on hold]

Say that $G'$ is a graph re-weighted from $G$ using the rule: $w'
(u, v) = w(u, v) − f(u) + f(v)$, where $f$ always produce the positive results for any nodes. Can we prove
that the shortest path ...

**5**

votes

**0**answers

54 views

### Are graphs with sparse $r$-balls necessarily sparse?

Let $G$ be an unweighted undirected graph with the following property:
For some integer $r$, for all nodes $v$, we have
$$\frac{\sum \limits_{u \in B(v, r)} \deg(u)}{|B(v, r)|} \le D$$
where $B(v, r) ...

**1**

vote

**0**answers

49 views

### Hashimoto Matrix (Non-backtracking operator) and the Graph Laplacian

The question is: how can we recover the graph Laplacian or its spectrum from the Hashimoto Matrix (also commonly called the Non-backtracking operator)?
To make the question as self-contained as ...

**7**

votes

**3**answers

174 views

### Graph homomorphisms and line graph

If $G,H$ are simple, undirected graphs, we say that they are in a hom(omorphism)-relation if there is a graph homomorphism from $G$ to $H$ or from $H$ to $G$.
For any graph $G$ let $L(G)$ denote its ...

**6**

votes

**1**answer

172 views

### Given k, what is the minimum n such that n choose n/2 is greater than k? [closed]

I'm not an expert in combinatorics, but it sometimes comes up in my research with students in computer science (which is already pretty far away from my speciality of abstract homotopy theory). I just ...

**7**

votes

**0**answers

103 views

### De Bruijn sequence inside De Bruijn sequence

A binary De Bruijn sequence of index $n$ is a circular sequence $S=a_1a_2\ldots a_{2^n}$, with $a_i∈\{0,1\}$, and such that each of the $2^n$ binary $n$-tuples occurs exactly once in $S$.
What is ...

**5**

votes

**1**answer

124 views

### How can I prove that these two graph coloring problems are polynomial time equivalent?

Given a graph $G(V,E)$. The standard $k$-coloring problem consists in finding a feasible coloring (no two adjacent nodes share the same color) of the nodes with $k$ colors. Let this problem be $P_1$.
...

**3**

votes

**0**answers

96 views

### Synonyms for “labeling” of a graph

In Preprint 1 we write numerical labels 0 or 1 at each vertex of a Dynkin diagram $D$. We call it a labeling of the graph (Dynkin diagram) $D$.
In Preprint 2 we consider an extended (affine) Dynkin ...

**3**

votes

**0**answers

53 views

### The degree/diameter problem for even girth graphs starting with upper bound

I posted this on stackexchange but due to a lack of response there I am posting here.
Let $G$ be a graph with girth $g$, minimal degree $\delta$, maximal degree $\Delta$, and diameter $D$. Define ...

**3**

votes

**1**answer

72 views

### Construction of planar embedding

I'm reading the following paper on universality considerations in VLSI circuits
http://www.computer.org/csdl/trans/tc/1981/02/06312176.pdf
In Theorem 2 On the second page it states there exists ...

**6**

votes

**3**answers

195 views

### Embedding planar graphs into the grid

I've seen the following lemma in a paper. The result is by Valiant.
A planar graph $G$ with maximum degree $4$ can be embedded in the plane using $O(|V|)$ area in such a way that its vertices are at ...

**6**

votes

**1**answer

116 views

### Refinement of Dirac's theorem on Hamiltonian graphs

Dirac's theorem states that if degree of each vertex of a graph $G=(V,E)$ is not less than $|V|/2$, then it has Hamiltonian cycle. It is less known, but still known and not so hard to prove (though I ...

**1**

vote

**2**answers

94 views

### SVD vs Fourier analysis for data.

Fourier analysis is useful for analysis in the frequency domain. SVD on the other hand is useful for analysis of data, and expressing noise in the data. I have a problem that needs extensive data ...

**-1**

votes

**1**answer

56 views

### Different graphs with the same open neighborhood hypergraph

For any set $X$ we let $[X]^2 = \big\{\{x,y\}: x\neq y \in X\big\}$.
Let $G=(V,E)$ be a simple, undirected graph. Its open neighborhood hypergraph $\mathcal{H}(G)$ has the same vertex set $V$ with a ...

**2**

votes

**1**answer

78 views

### First passage percolation for general graphs

There have been many questions about the behavior of first-passage percolation on specific graphs. In particular, it seems like cliques, grids, random graphs, and ladders are well-studied. But I can't ...

**1**

vote

**1**answer

55 views

### The existence of a specific partition of the edge set of $K_{2n}$

Let $n$ be an even positive integer and $K_{2n}$ be the complete graph on $2n$ vertices. There are $\dfrac{1}{2}{{2n}\choose n}={{2n-1}\choose n}$ subgraphs of $K_{2n}$ which is isomorphic to ...

**2**

votes

**0**answers

271 views

### Maps between graphs defined through laplacian operations

Edit: The views/answers ratio on this question tells me that it was too long. As such, I've stripped out examples and now ask the question in brief. For examples, please ask in the comments or look at ...

**5**

votes

**1**answer

81 views

### Above/below directed graph on cells of arrangement of lines

This question concerns the structure of a directed graph
built on the cells of an arrangement of lines.
My basic question is whether this graph has been
studied before, perhaps in another guise. I ...

**1**

vote

**2**answers

79 views

### Application of cospectral graphs

Cospectral graphs are graphs having same eigenvalues. Constructions of cospectral graph is an interesting question in graph theory. Now a days we use graph theory in different brunches of Sciences and ...

**1**

vote

**0**answers

65 views

### Computing the Edge Chromatic Polynomial of a graph

Is there a recursive formulae to compute the edge chromatic polynomial of a graph?
The following formulae is known for the vertex chromatic polynomial of a grapg $G$
$P(G,x)=P(G-uv, x)- P(G/uv,x)$ ...

**4**

votes

**1**answer

167 views

### Counting trees according to endpoints

Question: Is there a nice (or any) formula for the generating function
$$T(x,y) = \sum_{m,n} t_{m,n} x^my^n,$$
where $t_{m,n}$ is the number of trees with $m$ vertices and $n$ endpoints?
...

**1**

vote

**1**answer

134 views

### Number of nodes in a given distance in (random) regular graph

Given a d-regular graph $G=<V,E>$ (connected, unweighted & simple), and a node $v$.
denote all nodes with distance $k$ from $v$ $$L_k=\{u\in V : dis(v,u) = k\}$$
Let's call it "the k-th ...

**6**

votes

**2**answers

104 views

### Class of hypergraphs that are always the neighborhood hypergraph of some simple graph

Let $G=(V,E)$ be a graph. Its (open) neighborhood hypergraph $\mathcal{H}(G)$ has the same vertex set $V$ with a hyperedge for the (open) neighborhood of every vertex $v \in V$.
It seems that not ...

**2**

votes

**0**answers

31 views

### Is every 1-skeleton of a 4-tope Steinitzian?

Call a graph "polyhedral" if it is simple, planar, and 3-connected. For example, the 1-dimensional skeleton of every 3-dimensional convex polytope, regarded as a graph, is polyhedral. By a theoreom ...

**3**

votes

**1**answer

105 views

### Complexity of a very simple graph partitioning problem

The following problem seems like a very simple and natural one, but I am not familiar with any existing work on it; in particular I am hoping to prove it is NP hard:
Let $G$ be a complete weighted ...

**1**

vote

**0**answers

60 views

### Lp norm estimates for the inverse of the Laplacian on a graph

I am looking at a finite connected graph and I would like to know what is the best [i.e. largest] constant $\lambda_p$ in
$$
\sum_x f(x) =0 \implies \| \Delta^{-1} f\|_{\ell^p} \leq \lambda_p^{-1} ...

**1**

vote

**1**answer

67 views

### Similarity graph for continuous maps between Hausdorff spaces

Let $X, Y$ be topological spaces and $f,g: X\to Y$ continuous. Then we say that $f, g$ are similar if for all $V\subseteq Y$ open we have either
$f^{-1}(V) = g^{-1}(V) = \emptyset$, or
$f^{-1}(V) ...

**1**

vote

**1**answer

77 views

### Probability of having no cycles of fixed length in $d$-regular graphs

According to this paper, the probability that a random $d$-regular graph of order $n$ has no cycles of length $c_1,c_2,\ldots,c_t$ is $$P=\exp\left(-\sum_{i=1}^t\mu_i+o(1)\right)$$ as ...

**2**

votes

**3**answers

79 views

### Algorithm to determine isomorphism of 2 maximal planar graphs

I read on wikipedia that there are efficient algorithms to answer the question whether 2 (maximal) planar graphs F and G are isomorphic. However, after some (IMHO) substantial searching I don't seem ...

**4**

votes

**1**answer

139 views

### Partitioning ${\cal P}([[1,n]])$

In an analysis of the Jacobi method for the computation of the spectrum of a Hermitian matrix, I face the following problem.
Denote ${\cal P}_2(n)$ the set of doubletons $\{a,b\}$ in ...

**1**

vote

**0**answers

42 views

### Graph Theory for Dummies Book [migrated]

Does anyone have a good book on Graph Theory that will introduce me to some of the basic concepts without being so filled with terminology that it's hard to read? I have taken an introductory course ...

**3**

votes

**1**answer

94 views

### Method to construct a bipartite graph G' with 2n vertices from a graph G

I have seen mentioned in a talk an operation that takes a graph $G=(V,E)$ and constructs a new bipartite graph $G'=(V',E')$ such that $V' = V\times \{0,1\}$ and $E'=\{((i,1),(j,0)) : (i,j)\in E\} \cup ...

**5**

votes

**3**answers

216 views

### Generating (or availability of) large strongly regular graphs

Are there collections of already generated large strongly regular graphs available to download? By large I mean $n \geq 200$ where $n$ is the number of vertices. I found Ted Spence's page on srgs, ...

**1**

vote

**1**answer

64 views

### approximate diameter of polytopes in high dimensions

I just came across the following problem:
Let us consider the unit corner of the n-cube
$$
\Delta^n = \left\{(t_1,\cdots,t_n)\in\mathbb{R}^n\mid\sum_{i = 1}^{n}{t_i} \leq 1 \mbox{ and } t_i \ge 0 ...

**1**

vote

**1**answer

91 views

### Are constructive characterisations of k-regular (simple) graphs known?

By a constructive characterisation I mean a theorem giving a list of base graphs and a list of operations such that every graph in a given class is generated from the base graphs by applying some ...

**9**

votes

**1**answer

122 views

### What are some useful invariants for distinguishing between random graph models?

Quite a few probabilistic algorithms for generating random graphs exist in the literature, such as:
The Erdos-Renyi model
The Stochastic Block model
The Watts-Strogatz model
The Barabasi-Alber model
...

**2**

votes

**0**answers

56 views

### Non-existence of commutative rings with many nilpotent elements with some restrictions where matrix powers are efficient

At the moment can't find better reference than "Cycle Enumeration using Nilpotent Adjacency Matrices with Algorithm Runtime Comparisons"
though certainly there are others.
Consider the following ...

**2**

votes

**1**answer

58 views

### Maximum and minimum diameter of categorical graph product

Let $G_i$ be connected finite simple undirected graphs with diameter $d_i$ for $i=1,2$. Assume that $G_1\times G_2$ is connected. (Here $G_1\times G_2$ denotes the categorical product.)
In terms of ...

**1**

vote

**0**answers

82 views

### What characteristic of a graph depend on the vertex labeling?

Different labeling on a graph produces class of isomorphic graphs. Two isomorphic graphs possess similar characteristic such as connectivity, degree distribution of vertices, equality of spectrum and ...

**5**

votes

**0**answers

88 views

### “Edge Density” of Infinite Planar Graphs

Given an infinite planar graph $G$, let's denote by $\{H_1,H_2,\dots,H_m\}$ all the labeled graphs on $n$ vertices that appear as subgraphs of $G$. Also let
$$d_n=\frac{\sum_{i=1}^m \#E(H_i)}{nm}$$
...

**1**

vote

**0**answers

179 views

### “Graph Individualization”[ reference request] [closed]

Context: I am studying Weisfeiler Lehman method(WL method) and have clear idea about 1 and 2 dimensional WL method. I was wondering about the individualization process described below-
...

**2**

votes

**1**answer

177 views

### inequality with exponents

We are given a graph $G$, each vertex $v$ has an assigned value $\gamma_v\in [0,1]$, and it happens that for every $v$ we have $\gamma_v+\sum_{u\in \delta(v)} \gamma_u = 1$. Assume that $\sum_v ...

**2**

votes

**1**answer

128 views

### Embedding graphs into hyperbolic spaces

Do we know of a characterization as to when does a graph have a "good" embedding into a hyperbolic space? (And does having such an embedding have a spectral or wavelet analysis signature?)
I don't ...