The cycles tag has no wiki summary.

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### 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 ...

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331 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**

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**1**answer

258 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 ...

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**4**answers

687 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 ...

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vote

**1**answer

2k 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 ...

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**5**answers

456 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 $G$ then we can reduce the finding of a Hamiltonian cycle in $G$ to a Eurler your of $H$ ...

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**1**answer

143 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?

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**0**answers

98 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 ...

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**2**answers

234 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 ...

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**0**answers

139 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:
$rec/n: SK_1(X)/n \to \pi^{ab}_1(X)/n$
...

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**0**answers

251 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 ...

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**3**answers

642 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 ...

**1**

vote

**1**answer

250 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 ...

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**1**answer

215 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 ...

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vote

**1**answer

198 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 ...

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**1**answer

1k 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 ...

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votes

**2**answers

661 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 ...

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**3**answers

4k 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 ...

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**1**answer

443 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 ...