# Questions tagged [cycles]

The cycles tag has no usage guidance.

48
questions

5
votes

0
answers

142
views

### graphs where every cycle is a sum of triangles

I am studying a special kind of graphs, and I would like to know if they are studied in the literature and what they are called.
Let $G$ be a simple, finite, undirected, connected graph, with vertex ...

4
votes

1
answer

157
views

### Construction of graphs of high girth and chromatic number

Are there any concrete constructions of graphs of both high girth and chromatic number?
Of course there is the seminal paper of Erdős which proves the existence of such graphs via the probabilistic ...

2
votes

1
answer

128
views

### Number of endofunctions in [n] without fixed points with exactly k two-cycles

I need a (numerically) evaluable function for the number $N_{n,k}$ of endofunctions $f: [n] \rightarrow [n]$ without fixed points that have exactly $k$ two-cycles, where $[n] := \{1,\dotsc,n\}$. In ...

0
votes

0
answers

54
views

### Hamming distance globally and Euclidean distance locally to a cycle

Given a permutation matrix 'the question is to decide if there is a permutation matrix representing a cycle within Hamming distance $d$ from given matrix'. Is there an efficient algorithm for it?
...

1
vote

0
answers

87
views

### Sequences generated from commuted quaternions and general commuted linear transformations

Given a pair of non-commuting linear transformations, $A$ and $B$, define the "next
pair" in a sequence as $A*B$ and $B*A$. I am interested in finite cycles (i.e.,
the sequence eventually ...

3
votes

1
answer

144
views

### Probability permutation in turned to cycle

Let $M$ be a $0/1$ square matrix having one $1$ per row and column (permutation matrix).
If you permute the columns and rows independently what is the probability resulting permutation matrix is a ...

1
vote

0
answers

359
views

### Expected value of length of longest cycle in permutation

Let $n$ be a positive integer and let $S_n$ be the collection of permutations $\pi:\{1,\ldots,n\}\to \{1,\ldots,n\}$. For $\pi\in S_n$ let $\text{maxcyc}(\pi)$ denote the length of the longest cycle ...

1
vote

1
answer

216
views

### Unique bipartite perfect matchings and cycles?

Given a graph $G$ which is bipartite and balanced and has unique perfect matching let $G^{e}$ be $G$ without edge $e$. Let $G\cup G_{\pi,\pi'}$ be union of $G$ and $G_{\pi,\pi'}$ where $G_{\pi,\pi'}$ ...

2
votes

0
answers

154
views

### Abelian variety corresponding to a vector space

I would like to know what the following statement means:
"Let $B_t$ be the Abelian subvariety in $J_t$ corresponding to the $\mathbb{Q}$-vector subspace $H^1(C_t,\mathbb{Q})_{van}$ in the space $...

2
votes

2
answers

165
views

### Number of edge-disjoint cycles in a holey graph

Let $\Gamma$ be a connected graph with $H^1(\Gamma) \cong \mathbb{Z}^d$. Can we give a lower bound (preferably of the form $\gg d$) on the maximal number of edge-disjoint cycles one can find in $\...

2
votes

0
answers

224
views

### A fast algorithm for deciding if a given undirected graph contains a C4 subgraph

I'm looking for an algorithm for deciding if a given undirected graph G contains C4 as a sub graph, not necessarily induced. I'm not interested in finding such a cycle, if it exists.
I was told there ...

6
votes

3
answers

344
views

### Tree-width of graphs in which any two cycles touch

Let $G$ be a graph s.t. any two cycles $C_1, C_2 \subseteq G$ either have a common vertex or $G$ has an edge joining a vertex in $C_1$ to a vertex of $C_2$. Equivalently: for every cycle $C$ the graph ...

2
votes

1
answer

70
views

### Name for specific cycles in graphs

Is there an established name for cycles $C\subseteq G(V,E)$ with the property that
$$\lbrace u,v\rbrace\subseteq C\cap V\implies\mathrm{dist}_{|C}(u,v)\le \mathrm{dist}_{|G}(u,v)$$
I would be ...

2
votes

1
answer

65
views

### Algorithms for heaviest edge-disjoint cycle collection contained in graph's set of edges

given a biconnected symmetric graph with weighted edges,
what is the algorithmic complexity of determining a set of pairwise edge-disjoint cycles with maximal sum of edge weights if there are no other ...

3
votes

1
answer

269
views

### Bounding number of k-cycles in a graph

Fix any $k \geq 3$, and suppose I have a simple undirected graph $G=(V,E)$. I want a bound on the number of $k$ cycles in $G$ as a function of $|E|$. In particular, I would like to prove the following ...

2
votes

0
answers

95
views

### How many edges can be in an unbalanced bipartite graph of girth $>6$?

Let $G = (V, E)$ be a bipartite graph with $n, m$ nodes in its bipartition and girth (shortest cycle length) $>6$.
There is a simple counting argument called the Moore Bounds that gives
$$|E| = O\...

1
vote

1
answer

234
views

### Matrix logarithm for d-dimensional cyclic permutation matrix

I want to find the matrix $\hat{H}_d$ which, when exponentiated, leads to a d-dimensional cyclic permutation transformation matrix.
I have solutions for d=2:
$$
\hat{U}_2 =\left( \begin{matrix}
...

7
votes

1
answer

345
views

### Strong tournaments

Let $T$ be a strong tournament, and let $N=v_1v_2 \cdots v_n$ be an enumeration of $V(T)$. Let $C$ be a circuit in $T$. We define $i_N(C)=|\{(v_i,v_j) \in E(C); i>j\}|$. Suppose that $N$ is chosen ...

6
votes

0
answers

173
views

### A question about dominating circuits in cubic graphs

Let $G$ be a 3-connected cubic graph with a dominating circuit $C$, that is, a circuit such that all edges in $G$ have at least one endvertex in $C$. Let $D$ be another circuit and let the symmetric ...

3
votes

1
answer

225
views

### Cycle Structure of a Permutation Based on the Binary Representation

This is a question I posted on math.stackexchange.com before but never got an answer. I am cross-posting it here.
Define a permutation $\sigma$ on the set $X=\{1,2,...,n\}$, $n$ is a natural number ...

1
vote

2
answers

188
views

### Extremal density of a graph without a non-backtracking $2k$-cycle

The current best bound for the maximum possible density of an $n$-node graph with girth (shortest cycle length) $>2k$ is of the form
$$ex(n \ \mid \ C_{\le 2k}) = O(n^{1 + 1/k}),$$
while the ...

3
votes

1
answer

590
views

### Pull-back of algebraic cycles

Since today is the Chow-variety day, I'm going to ask my question here.
Suppose I have a smooth projective variety $X$ over a field of characteristic zero, and a smooth hyperplane $p: H\...

7
votes

2
answers

316
views

### Minimum covers of complete graphs by $4$-cycles

I am interested in coverings of the (edge set of the) complete graph $K_n$ by cycles of length $4$. It is clear that such coverings exist for each $n \ge 4$. I need to find the minimum number of $4$-...

8
votes

4
answers

328
views

### Iteration cycles of Z_n weights in path graphs: Why cycles of length 182 for a 6-node path?

Assign to the $n$ nodes of a path graph vertex weights
forming a permutation of $(0,\ldots,n{-}1)$.
Now iterate the following update repeatedly:
Each node sums the weights of its neighbors, and that ...

6
votes

0
answers

101
views

### Localizing Bondy's metaconjecture on hamiltonicity

Definitions:
Let $G$ be a graph on $n$ vertices. $G$ is Hamiltonian provided $G$ has a cycle of length $n$. $G$ is pancyclic provided $G$ has a cycle of length $\ell$ for every $3 \leq \ell \leq n$.
...

2
votes

0
answers

299
views

### On symmetric difference of $k$-partite perfect matchings

Given a bipartite graph we know that symmetric difference of any two perfect matchings is union of even cycles.
Conversely when is it true that every union of even cycles comes from symmetric ...

1
vote

1
answer

126
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 $n\rightarrow\...

2
votes

2
answers

946
views

### Cycle class map in smooth quasi-projective varieties

Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$ and $Z$ be a closed subvariety of codimension $k$.
Q1. How to define a cycle class $[Z]\in H^k(X,\Omega_X^{k})$ ?
Q2. More general, ...

1
vote

1
answer

163
views

### Does every connected vertex transitive graph on $n$ vertices (except for $C_n$) have minimum feedback vertex set of size $\Omega(n)$?

Feedback vertex set is a set of vertices whose removal leaves an acyclic graph.
It is known that every vertex transitive graph on $n$ vertices has minimum vertex cover of size $\Omega(n)$. It is also ...

7
votes

2
answers

7k
views

### How many simple cycles can a graph with $n$ vertices and $m$ edges have?

I am mainly interested in the smallest number of simple cycles a graph with $n$ vertices and $m$ edges must have.
For example, if $m\le n-1$, this number is $0$, then if $n\le m \le 3(n-1)/2$, it is $...

4
votes

2
answers

446
views

### Do graphs with large number of cycles always contain large necklace minor?

Let "$k$-necklace" denote the (multi)graph obtained from a cycle of length $k$ by duplicating every edge.
Note that the number of cycles in $k$-necklace is at least $2^k.$
Question : Suppose a ...

4
votes

1
answer

568
views

### Analytic continuation of a multiple contour integral

Let $W(t_1,\dotsc,t_n)$ a holomorphic function on some connected open set $U$ of $\mathbb C^n$.
Let $\mathbf t^{(0)}$ a point of $U$.
Assume that there exists a cycle $\gamma$ in $\mathbb C^m$ and a ...

2
votes

4
answers

5k
views

### Counting simple 4-cycles in an undirected graph [closed]

I'm looking for an algorithm which just counts the number of simple and distinct 4-cycles in an undirected graph labelled with integer keys. I don't need it to be optimal because I only have to use it ...

6
votes

6
answers

29k
views

### Calculating Pisano periods for any integer

I recently stumbled across this SPOJ question:
http://www.spoj.com/problems/PISANO/ The question is simple. Calculate the Pisano period of a number. After I researched my way through the web, I found ...

7
votes

6
answers

3k
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 $H$, then we can reduce the problem of finding a Hamiltonian cycle in $G$ to finding an ...

3
votes

1
answer

189
views

### On cycles in self-centered graphs

Let $G$ be (connected) self-centered graph, i.e. $r(G)=d(G)=m<\infty$.
My question is following
Does $G$ always contains $C_{2m}$ or $C_{2m+1}$ as a subgraph?

0
votes

0
answers

131
views

### Notation for substructure, especially for permutations?

Is there a standard notation that expresses substructure?
The specific case that I care about is the following:
Suppose $\sigma,\tau$ are permutations such that $$\sigma(x)\not=x\implies \sigma(x)=\...

1
vote

2
answers

348
views

### Ihara zeta function (graph theory) coefficients using a line graph [closed]

I'VE COMPLETELY REVISED MY QUESTION
I wish to take a simple undirected graph (i.e. the complete graph K_4)
Arbitrarily direct said graph, and then create a line graph from the directed version of ...

4
votes

0
answers

222
views

### Reciprocity Map and Cycle Class Map

This might be a very naive question but here it goes. Let X be a smooth variety of dimension d over a p-adic field. We have the n part of the rerciprocity map:
$rec/n: SK_1(X)/n \to \pi^{ab}_1(X)/n$
...

4
votes

0
answers

598
views

### Expected number of components with multiple cycles in a subgraph of a square lattice

Short version
Is there an understanding of the emergence and subsequent disappearance of components with zero, one, or more cycles in a random subgraph of a square or cubic lattice, as the edge-...

7
votes

3
answers

876
views

### 2-cycle of K3 surface

Hi there,
I want to ask about the 2-cycle of K3 surface.
As we know, its betti number $b_2$=22, so there will be 22 2-cycle generators.
Is there any topological way to figure out such cycles direct?...

2
votes

2
answers

520
views

### Need input on a potentially NP-hard maximal edge-weighted multi-cycle graph

I've posted a question on Stack Overflow regarding a seemingly NP-hard problem on maximization of weighted cycles in a graph problem.
One of the respondents cited Professor David Speyer's Math ...

0
votes

1
answer

615
views

### finding missing edge in DAG which, when added, would create the longest cycle

Hey all,
Not sure if this is a math problem or an algorithm problem - but hoping it has a math style answer.
If I have a directed graph I can find all the closed loops - easy. (Actually not at all ...

1
vote

1
answer

335
views

### Definition of convex cycles

Consider the following definition.
Let $C$ be a cycle of a simple graph $G$. We say that $C$ is convex if for any pair of distinct vertices $u,v \in V(C)$ $$ d_C(u,v) < d_{G-C}(u,v).$$
Is there ...

4
votes

1
answer

2k
views

### How many edge-disjoint cycles of length 3 are in the complete graph?

A couple of questions related to edge-disjoint cycles.
Let $K_n = (V,E)$ be the complete graph on $|V|=n$ nodes. Two cycles are 'edge disjoint' if they do not share any edges.
What is the size of ...

8
votes

2
answers

3k
views

### Ramification divisor associated to a cover of a regular scheme

Let $S$ be the spectrum of $\mathbf{Z}$ or the spectrum of an algebraically closed field. (Actually, one can take $S$ to be any noetherian integral regular scheme.)
Let $f:X\longrightarrow Y$ be a ...

2
votes

3
answers

14k
views

### Cycle of length 4 in an undirected graph

Can anyone give me a hint for an algorithm to find a simple cycle of length 4 (4 edges and 4 vertices that is) in an undirected graph, given as an adjacency list? It needs to use $O(v^3)$ operations (...

3
votes

1
answer

478
views

### Compute number vertex disjoint cycles in graph surrounding a face

Hi all,
If anyone has insight into the following variant of the classic problem of packing vertex-disjoint cycle into graphs I would be interested.
Given a finite undirected graph $G$ embedded in $...