Problem 2 of Assignment 1 is not true as stated.  As mentioned in the problem and in class, the point was to fill in the missing step in the proof of the planar separator theorem given by Alon, Seymour and Thomas.  The precise version they need is the following: 

Let G be a planar graph whose inner faces are triangulated.  Let the outer face be (in order) s_1, s_2, . . s_k = t_k, t_{k-1} . . t_1 = s_1.   i.e. the outer face has 2k-2 vertices.  Either there exist k vertex disjoint paths p_i joining s_i to t_i for i=1, . . k 
OR there exists a path of less than k vertices from s_1 to s_k. 

In particular, the outer face ONLY contains the s_i's and t_i's, and the path is allowed to contain s_i and t_i vertices. 

The version I stated in the problem is more general in allowing extra vertices on the outer face, and requiring the path to separate the s_i's from the t_i's.  That general version is not true.

Please regard this problem as a more open-ended one:  Try to prove the precise version that's needed for the separator theorem.  Why is the result false in the more general version?

Let me know if you have further questions.  Thank you to Abbas Mehrabian for pointing this out.