Abstract:
A vertex-transitive but not edge-transitive graph Γ is called locally bi-2-transitive if the stabiliser S in the full automorphism group of Γ of every vertex v of Γ has two orbits of equal size on the neighbourhood of v, and S acts 2-transitively on each of these two orbits. Also a graph is called cycle-regular if the number of cycles of a given length passing through a given edge in the graph is a constant, and a graph with girth g is called edge-girth-regular if the number of cycles of length g passing through any edge in the graph is a constant. In this paper, we prove that a graph of girth 3 is edge-girth-regular and locally bi-2-transitive if and only if Γ is the line graph of a semi-symmetric locally 3-transitive graph. Then as an application, we prove that every tetravalent edge-girth-regular locally bi-2-transitive graph of girth 3 is cycle-regular. This shows that vertex-transitive cycle-regular graphs need not to be edge-transitive, and hence resolves the problem posed by Fouquet and Hahn at the end of their paper ‘Cycle regular graphs need not be transitive’, in Discrete Appl. Math. 113 (2001) 261–264.
