Distinguishing Maps

Abstract

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 $D(M)=D({\rm Aut}(M),V)$, where ${\rm Aut(M)}$ is the automorphism group of $M$; if $M$ is orientable, define $D^+(M)$ similarly, using only orientation-preserving automorphisms. It is immediate that $D(M)\leq 4$ and $D^+(M)\leq 3$. 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 $\chi_D(M)$, where adjacent vertices get different colors. We show $\chi_D(M)\leq \chi(M)+3$ with equality in only finitely many cases, where $\chi(M)$ is the chromatic number of the graph underlying $M$. We also show that $\chi_D(M)\leq 6$ 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.

Endomorphism Breaking in Graphs

“…The aim of this paper is the presentation of fundamental results for the endomorphism distinguishing number, and of open problems. In particular, we extend the Motion Lemma of Russell and Sundaram [14] to endomorphisms, present endomorphism motion conjectures that generalize the Infinite Motion Conjecture of Tom Tucker [16] and the Motion Conjecture of [4], prove the validity of special cases, and support the conjectures by examples.…”

“…This concept has spawned numerous papers, mostly on finite graphs. But countable infinite graphs have also been investigated with respect to the distinguishing number; see [9], [15], [16], and [17]. For graphs of higher cardinality compare [10].…”

Breaking graph symmetries by edge colourings

“…The same is true for infinite graphs whose automorphism group is finite. Tucker [15] conjectured, that an analogous result holds for locally finite graphs with infinite automorphism group.…”

“…Conjecture 1 (Infinite motion conjecture [15]). Let G be a locally finite, connected graph and assume that every automorphism of G moves infinitely many vertices.…”

Local Finiteness, Distinguishing Numbers, and Tucker's Conjecture

“…For infinite graphs one of the most intriguing questions is whether or not the following conjecture of Tucker [12] is true. Conjecture 1.1.…”