In this paper we discuss an unsolved problem in [1]: Determine which simple graph G has exactly one cycle of each length l, 3 ~< l ~< v (where v is the number of the vertices of G). We call a graph with this property a uniquely pancyclic graph (UPC-graph). We solve this problem under the condition: G is an outerplanar graph. We determine all UPC-graphs each of which contains v + m edges for m ~< 3. We also conjecture that none of the graphs, each of which contains v + m edges for m/> 4, is a… CONTINUE READING