A classification of pentavalent arc-transitive bicirculants

Abstract

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 $$\mathrm{Cay}(D_{2n},\{b,ba,ba^{r+1},ba^{r^2+r+1},ba^{r^3+r^2+r+1}\})$$Cay(D2n,{b,ba,bar+1,bar2+r+1,bar3+r2+r+1}) on the dihedral group $$D_{2n}=\langle a,b\mid a^n=b^2=baba=1 \rangle $$D2n=?a,b?an=b2=baba=1?, where $$r\in \mathbb {Z}_n^*$$r?Zn? such that $$r^4+r^3+r^2+r+1 \equiv 0 \pmod {n}$$r4+r3+r2+r+1?0(modn).