Vrftsjtryk

So I found some proofs on any two vertices would lie on a cycle, but stuck on dealing with edges. We can say any two edges are connected, but does that just imply they will be on a common cycle?

Let $G = (V, E)$ graph, $|V| ge 3$. Suppose that there exists an edge $e = (u, v) in E$ such that $e$ does not lie on a cycle. Then every path from $u$ to $v$ must contain $e$ (1). If $u$ or $v$ are a leaf, then removing the vertex that connects the leaf from the rest of the graph makes the leaf disconnected (!). Otherwise, define the following partition of $G$ without $e$:

$$G_u = (V_u, E_u) quad quad G_v = (V_v, E_v)$$

where $V_u = {w in V |$ there is a path from $w$ to $u$ that does not contain $e}$, $E_u = {(w, t) in E | w, t in V_u}$ and $V_v$, $E_v$ are defined analogously (2). If we remove $u$ or $v$ from the graph, then we are also removing $e$. Note, however, that if we remove $e$, then there is no path to go from any $w in G_u$ to any $t in G_v$. In other words, the graph becomes disconnected (!).

Notes:

(1) Suppose that there exists a path from $u$ to $v$ that does not contain $e$. Then we can join this path to $e$ to form a cycle, but by doing so we are showing that $e$ lies on a cycle (!).

(2) It is easy to show that $G_u$ and $G_v$ are disjoint.