Cycle graph

Example

Let $n \ge 3$ be an integer. The cycle graph $C_n$ is the graph 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 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.

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.