Distinguishing index of maps

Abstract The distinguishing number of a group A acting on a finite set Ω , denoted by D ( A , Ω ) , is the least k such that there is a  k -coloring of Ω which is preserved only by elements of A fixing all points in Ω . For a map M , also called a cellular graph embedding or ribbon graph, the action of Aut ( M ) on the vertex set V gives the distinguishing number D ( M ) . It is known that D ( M ) ≤ 2 whenever | V | > 10 . The action of Aut ( M ) on the edge set E gives the distinguishing index D ′ ( M ) , which has not been studied before. It is shown that the only maps M with D ′ ( M ) > 2 are the following: the tetrahedron; the maps in the sphere with underlying graphs C n , or K 1 , n for n = 3 , 4 , 5 ; a map in the projective plane with underlying graph C 4 ; two one-vertex maps with 4 or 5 edges; one two-vertex map with 4 edges; or any map obtained from these maps using duality or Petrie duality. There are 39 maps in all.