Let's say that a graph G has the "unique path property" if there's a unique shortest path connecting any two vertices of G. Show that G has the unique path property if and only if every even cycle of G has at least two internal edges.
posted by Lionel Levine at 5:58 PM

Pretty sure these are true but I'll make them PODASIPS just in case! Prove or disprove:

1. S^n cannot be covered by fewer than n+2 open hemispheres.
2. If S^n = union of H_alpha is any covering of the sphere by open hemispheres, there is a subcover S^n = union of H_alpha_i consisting of n+2 of them.
posted by Lionel Levine at 5:49 PM