Currently I am reading the first edition "Introduction to Graph Theory" by Douglas B. West.
Exercise 1.1.9 states the following:
1.1.9. Suppose that $G$ is a simple graph having no vertex of degree $0$ and no induced subgraph with three edges. Prove that $G$ has at most four vertices.
Counter-example: Let $V(G) = ${$ 1,2,3,4,5 $}$ $ and $E(G) = ${$ 12,23,45 $}$ $.
Is this counter-example correct ?
EDIT:
Came up with this approach but I'm stuck.
Proof. Because the degree of every vertex in $G$ has degree of at least 1 then
$$ \sum_{v \in V}^{} \text{deg}(v) \geq |V| $$
and recall that half the sum of all degrees in $G$ is the size of the edge set, thus
$$ |E|\geq \frac{1}{2} |V| $$
Now, if $G$ does not contain any induced subgraph $H$ with three edges, that implies there is no 4-vertex or 5-vertex subset of $V(G)$ for which the induced subgraph under that subset contains three edges. Because $|E| \geq 3$ for $|V| \geq 5$ then $G$ must contain a three edge subgraph (and im stuck here)
Your Counter-Example almost contains the Proof !
When we have graph $G$ with $5$ or more vertices , then a sub-graph ( which might be $G$ itself ) will contain $5$ vertices , which will look like your Example.
Let all the vertices be $0$ Degree , unconnected to other vertices.
We want the Case where there are no $0$ Degree vertices , while we have $5$ such $0$ Degree vertices. Hence we can connect ( arbitrarily ) vertices $1$ & $2$ to get $(1,2)$.
We still have $3$ vertices with $0$ Degree , with $1$ Edge.
Adding a new Edge either will reduce that count by $1$ ( eg $(1,3)$ ) or will reduce that count by $2$ ( eg $(3,4)$ ) , leaving us with atleast $1$ vertex with $0$ Degree.
Adding a third Edge will give a subgraph with 3 Edges , while not adding a third Edge will give a graph with at least $1$ unwanted $0$ Degree vertex.
Either way , the Conclusion holds !
At the Point where we have 3 Edges , adding more Edges might eliminate all the $0$ Degree vertices , though it can not eliminate the currently existing subgraph having $3$ Edges.
Maximal Example : $E(G)=\{1,2,3,4\}$ with $V(G)=\{(1,2),(3,4)\}$