Distinguishing Infinite Graphs

Abstract

The distinguishing number $D(G)$ of a graph $G$ is the least cardinal number $\aleph$ such that $G$ has a labeling with $\aleph$ labels that is only preserved by the trivial automorphism. We show that the distinguishing number of the countable random graph is two, that tree-like graphs with not more than continuum many vertices have distinguishing number two, and determine the distinguishing number of many classes of infinite Cartesian products. For instance, $D(Q_{n}) = 2$, where $Q_{n}$ is the infinite hypercube of dimension ${n}$.

Base size, metric dimension and other invariants of groups and graphs

“…In their 2007 paper [67], Imrich et al considered the countable random graph R (also known as the Rado graph: see [33, Section 5.1]), which has a large automorphism group, and proved that Aut(R) has distinguishing number 2. Their result has been generalized in a recent paper of Laflamme et al [74], where they considered a broader class of structures.…”

Distinguishing graphs with infinite motion and nonlinear growth

“…Notice that homogeneous trees of degree d > 2, that is, infinite trees where every vertex has the same degree d, have exponential growth. For the distinguishability of such trees and tree-like graphs, see [16] and [9].…”

“…This seminal concept spawned many papers on finite and infinite graphs. We are mainly interested in infinite, locally finite, connected graphs of polynomial growth, see [8], [15], [13], and in graphs of higher cardinality, see [9], [11]. In particular, there is one conjecture on which we focus our attention, the Infinite Motion Conjecture of Tom Tucker.…”

Endomorphism Breaking in Graphs

“…We conjecture that this result can be extended to uncountable trees. One does need a lower bound on the minimum degree though, see [10]. As we already noted, the fact that D e (T ) = 2, together with the observations that |End(T )| = c and m e (T ) = ℵ 0 , supports the Endomorphism Motion Conjecture.…”

“…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].…”