Abstract
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⩽ ν (where ν 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 ν + m edges for m⩽3. We also conjecture that none of the graphs, each of which contains ν + m edges for m⩾4, is a UPC-graph, and we prove that this conjecture for m=4 is true.
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