Please mark the following 5 questions, 2 marks each.  You can give 1/2 marks.

Reminder to put a comment box on the cover page stating which questions were marked.

11.4 # 19.  The answer 10 is in the book.  The marks
     are all for justification of the solution.
     1 mark for using the degree sum formula to show that |V|=8  
     1 mark for using Euler's formula |V|-|E|+|F|=2 to show |F|=10

12.1 # 10.
    "If a connected undirected graph G=(V,E) has 30 edges, what
     is the maximum value for |V|?"
    1 mark for |V|=31 and 1 mark for justification.

Instructor questions

Q1  1 mark for the planar drawing with straight lines for edges
    1 mark for the planar dual
    Note, the graph P is a pyramid graph and it's dual P* = P

Q5  1 mark for writing the converse of Theorem 11.8 [see solutions]
    1 mark for giving a counter example

Q9  Let T be a tree with >= 2 vertices.
    Prove that removing an edge in T disconnects T.

    1.5 marks for a correct proof
    0.5 marks for writing

    Here is a third likely proof.
    
    Let T=(V,E).  In T we have |V|=|E|+1.
    If we remove an edge e then T-e there is one less edge so that |V| \ne |E|+1
    does not hold so T-e is not a tree.  Since T is a tree, T is acyclic.
    Removing an edge cannot create a cycle so T-e is acyclic.  Since T-e is not
    a tree, and T-e is acyclic then it must be disconnected.
