A note on the girth‐doubling construction for polygonal graphs

A near‐polygonal graph is a graph Γ which has a set of m ‐cycles for some positive integer m such that each 2‐path of Γ is contained in exactly one cycle in . If m is the girth of Γ then the graph is called polygonal. Given a polygonal graph Γ of valency r and girth m , Archdeacon and Perkel proved the existence of a polygonal graph Γ 2 of valency r and girth 2 m . We will show that this construction can be extended to one that yields a polygonal graph Γ 3 of valency r and girth 3 m , but that making the cycles any longer with this construction does not yield a polygonal graph. We also show that if Aut(Γ) is 2‐arc transitive, so is Aut(Γ k ) for k = 2, 3. © 2010 Wiley Periodicals, Inc. J Graph Theory 68: 246‐254, 2011