A classification of pentavalent arc-transitive bicirculants

A bicirculant is a graph admitting an automorphism with two cycles of equal length in its cycle decomposition. A graph is said to be arc-transitive if its automorphism group acts transitively on the set of its arcs. All cubic and tetravalent arc-transitive bicirculants are known, and this paper gives a complete classification of connected pentavalent arc-transitive bicirculants. In particular, it is shown that, with the exception of seven particular graphs, a connected pentavalent bicirculant is arc-transitive if and only if it is isomorphic to a Cayley graph Cay(D-2n, {b, ba, ba(r+1), ba(r2+r+1), ba(r3+r2+r+1)}) on the dihedral group D-2n = < a, b | a(n) = b(2) = baba = 1 >, where r is an element of Z(n)* such that r(4) + r(3) + r(2) + r + 1 = 0 (mod n).