A and B were good for the most part.  There were several people
who gave a lot of redundancy when asked for the cycles--they didn't recognize that, say, p,q,r,p and
r,p,q,r represent the same cycle.

for B, part e, many people said yes and gave a path, but didn't explain in words why demonstrating the
path was sufficient.

C most people understood this.

D Some people wrote that if u_i = u_j for some i<j, then the part of
the path from u_i to u_j forms a cycle.  This is false in general,
as there can be other repeating vertices in between.  What is true
is that this part of the path forms a closed path.  Also, some people
wrote that "removing" this part of the path would result in a simple path, which
is false in general (since there can still be repeating vertices left).
Some people wrote vaguely about "removing loops" to get a simple path, which need to be
made more precise (e.g., by working with the path notation u_1, u_2, ..., u_{p+1}).

E a handful of people didn't split this into the "if" and "only if" directions.  Several people said
in word why it is true, but did not construct particular paths (either between specific vertices or
one containing all the vertices).  A handful of people picked up on the fact that one direction was in
your notes, and this helped them construct their arguments for the other directions.  If people are
feeling unsure about how to write proofs, I think mimicry is a great way to learn.  They have your
notes and can see the proofs in there are structured and can mimic the approaches taken.

F:  several people had difficulties with finding the maximum number of arcs for c & d, that is,
counting.  Perhaps some students lack combinatorics backgrounds.  For the minimum number of arcs,
several people put either n or n-1 for both c and d, not realizing they will actually be different.

G:  for a and b, several people had difficulties with writing the paths in general.  They would put
u1, u5, u6, ... instead of u1, u2, \dots, ui, u(i+5), ...
c confused a lot of people.  They somehow thought that when the arc was removed, the path should have
the same property (i and i+4 from the beginning).
d some people drew graphs but didn't demonstrate a path that exhibited the desired property

H:  several people did not attempt this at all--they struggled with how to conceptualize the problem.
A handful of people had bids as nodes and directed arcs connecting bids that are consecutive with
respect to time.