(a)
METHOD 1
Sum of degrees of vertices = \( 3 + 5 + 5 + 5 + 4 + 4 = 26 \)
Number of edges \( e = 13 \)
The sum is equal to twice the number of edges which verifies the handshaking lemma
METHOD 2
Degrees of vertices = 3, 5, 5, 5, 4, 4
There are 4 vertices of odd degree
There is an even number of vertices of odd degree which verifies the handshaking lemma
Marks: (M1) for listing degrees, (A1) for sum or odd vertex count, (A1) for verification
Result: \( G \) satisfies the handshaking lemma [3]

(b)
If planar then \( e \leq 3v – 6 \)
\( e = 13 \), \( v = 6 \)
Inequality not satisfied
Therefore \( G \) is not planar
Marks: (M1) for stating formula, (A1) for substituting \( e = 13 \), \( v = 6 \), (A1) for conclusion
Result: \( G \) is not planar [3]

(c)
There are vertices of odd degree
Hence it does not contain an Eulerian circuit
Marks: (A1) for stating no Eulerian circuit, (R1) for reason
Result: No Eulerian circuit due to odd-degree vertices [2]