Isomorphisms of bi-Cayley graphs on Dihedral groups.

For a group G and a subset S of G the bi-Cayley graph BCay (G , S) of G with respect to S is the bipartite graph with vertex set G × { 0 , 1 } and edge set { { (x , 0) , (s x , 1) } | x ∈ G , s ∈ S }. A bi-Cayley graph BCay (G , S) is called a BCI-graph if for any bi-Cayley graph BCay (G , T) , BCay (G , S) ≅ BCay (G , T) implies that T = g S α for some g ∈ G and α ∈ Aut (G). A group G is called a m -BCI-group if all bi-Cayley graphs of G with valency at most m are BCI-graphs. In this paper, we characterize the m -BCI dihedral groups for m ≤ 3. Also, we show that the dihedral group D 2 p (p is prime) is a 4 -BCI-group. [ABSTRACT FROM AUTHOR]