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
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Sources

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