Characteristics of Cycle Graph

Theorem

Let $G = \struct {V, E}$ be an (undirected) graph whose order is greater than $2$.

Then $G$ is a cycle graph if and only if:

$G$ is connected

every vertex of $G$ is adjacent to $2$ other vertices

every edge of $G$ is adjacent to $2$ other edges.

Proof

Recall that a cycle is a closed walk with the properties:

all its edges are distinct

all its vertices (except for the start and end) are distinct.

From Cycle Graph is Connected, $G$ is connected.

From Cycle Graph is 2-Regular, $G$ is $2$-regular.

Hence every vertex of $G$ is incident with $2$ edges.

This needs considerable tedious hard slog to complete it. In particular: I hate trying to prove obvious things To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Chapter $1$: Mathematical Models: $\S 1.3$: Graphs: Problem $23$