---
tag: 0QQ0F12962
kind: example
name: Cycle graph
msc: [05C38]
instance_of: [00C0BR7D7H]
sources:
  - url: https://en.wikipedia.org/wiki/Cycle_graph
    locator: lead section
same_as:
  wikidata: Q622506
  wikipedia: Cycle graph
---

## Example

Let $n \ge 3$ be an integer.
The cycle graph $C_n$ is the [[graph|0JD5C82QMB]] with vertex set $\mathbb{Z}/n\mathbb{Z}$ in which $i$ and $j$ are adjacent if and only if $j - i = \pm 1$.
Its edges are $\{0, 1\}, \{1, 2\}, \dots, \{n-2, n-1\}$ and $\{n-1, 0\}$.

## Remarks

$C_n$ has $n$ vertices and $n$ edges, and it is $2$-[[regular|0KA4S8R39M]] and connected.
$C_3$ is the complete graph $K_3$.
The chromatic number of $C_n$ is $2$ for even $n$ and $3$ for odd $n$. The odd cycles are the obstruction in the theorem that [[a graph is bipartite iff it has no odd cycle|01R1HPEK6S]].

## Explanations

### L1 human

A cycle graph is a ring of $n$ vertices, each joined to the next one and the last joined to the first.
It looks like a polygon with $n$ corners.