Distinguishing Maps

The distinguishing number of a group A acting faithfully on a set X, denoted D(A,X), is the least number of colors needed to color the elements of X so that no nonidentity element of A preserves the coloring. Given a map M (an embedding of a graph in a closed surface) with vertex set V and without loops or multiples edges, let D(M) = D(Aut(M), V ), where Aut(M) is the automorphism group of M ; if M is orientable, define D+(M) similarly, using only orientation-preserving automorphisms. It is immediate that D(M) ≤ 4 and D+(M) ≤ 3. We use Russell and Sundaram’s Motion Lemma to show that there are only finitely many maps M with D(M) > 2. We show that if a group A of automorphisms of a graph G fixes no edges, then D(A,V ) = 2, with five exceptions. That result is used to find the four maps with D+(M) = 3. We also consider the distinguishing chromatic number χD(M), where adjacent vertices get different colors. We show χD(M) ≤ χ(M) + 3 with equality in only finitely many cases, where χ(M) is the chromatic number of the graph underlying M . We also show that χD(M) ≤ 6 for planar maps, answering a question of Collins and Trenk. Finally, we discuss the implications for general group actions and give numerous problems for further study.