Classification of 2-arc-transitive dihedrants

A complete classification of 2-arc-transitive dihedrants, that is, Cayley graphs of dihedral groups is given, thus completing the study of these graphs initiated by the third author in [D. Marušič, On 2-arc-transitivity of Cayley graphs, J. Combin. Theory Ser. B 87 (2003) 162–196]. The list consists of the following graphs:(i) C 2 n , n ⩾ 3 ; te graphs K 2 n , n ⩾ 3 ; te bipartite graphs K n , n , n ⩾ 3 ; te bipartite graphs minus a matching K n , n − n K 2 , n ⩾ 3 ; nce and nonincidence graphs B ( H 11 ) and B ′ ( H 11 ) of the Hadamard design on 11 points; nce and nonincidence graphs B ( PG ( d , q ) ) and B ′ ( PG ( d , q ) ) , with d ⩾ 2 and q a prime power, of projective spaces; infinite family of regular Z d -covers K q + 1 2 d of K q + 1 , q + 1 − ( q + 1 ) K 2 , where q ⩾ 3 is an odd prime power and d is a divisor of q − 1 2 and q − 1 , respectively, depending on whether q ≡ 1 ( mod 4 ) or q ≡ 3 ( mod 4 ) , obtained by identifying the vertex set of the base graph with two copies of the projective line PG ( 1 , q ) , where the missing matching consists of all pairs of the form [ i , i ′ ] , i ∈ PG ( 1 , q ) , and the edge [ i , j ′ ] carries trivial voltage if i = ∞ or j = ∞ , and carries voltage h ¯ ∈ Z d , the residue class of h ∈ Z , if and only if i − j = θ h , where θ generates the multiplicative group F q ∗ of the Galois field F q .