4 The connected planar graph \(P\) has the adjacency matrix
| \cline { 2 - 6 }
\multicolumn{1}{c|}{} | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) |
| \(A\) | 0 | 1 | 1 | 0 | 1 |
| \(B\) | 1 | 0 | 1 | 0 | 1 |
| \(C\) | 1 | 1 | 0 | 1 | 1 |
| \(D\) | 0 | 0 | 1 | 0 | 1 |
| \(E\) | 1 | 1 | 1 | 1 | 0 |
4
- Draw \(P\)
4
- Using Euler's formula for connected planar graphs, show that \(P\) has exactly 5 faces.
4
- Ore's theorem states that a simple graph with \(n\) vertices is Hamiltonian if, for every pair of vertices \(X\) and \(Y\) which are not adjacent,
$$\text { degree of } X + \text { degree of } Y \geq n$$
where \(n \geq 3\)
Using Ore's theorem, prove that the graph \(P\) is Hamiltonian.
Fully justify your answer.