The Distinguishing Index of Infinite Graphs

Abstract

The distinguishing index $D^\prime(G)$ of a graph $G$ is the least cardinal $d$ such that $G$ has an edge colouring with $d$ colours that is only preserved by the trivial automorphism. This is similar to the notion of the distinguishing number $D(G)$ of a graph $G$, which is defined with respect to vertex colourings.We derive several bounds for infinite graphs, in particular, we prove the general bound $D^\prime(G)\leq\Delta(G)$ for an arbitrary infinite graph. Nonetheless, the distinguishing index is at most two for many countable graphs, also for the infinite random graph and for uncountable tree-like graphs.We also investigate the concept of the motion of edges and its relationship with the Infinite Motion Lemma.

The optimal general upper bound for the distinguishing index of infinite graphs

“…We need one more definition and a theorem proved by Broere and Pilśniak in . Let be a vertex in .…”

Bounds for Distinguishing Invariants of Infinite Graphs

“…They also introduced the distinguishing chromatic index χ D (G) as the minimum number of colours needed to properly colour the edges of G such that this edge colouring is only preserved by the trivial automorphism. Furthermore, they showed that χ D (G) ∆(G) + 1 except for C 4 , K 4 , C 6 and K 3,3 . Therefore, except for these four small graphs, we have χ D (G) χ (G) + 1, a result which is also true for infinite graphs, as we show in Section 4.…”

“…In 2015, Broere and Pilśniak [3] considered D (G) for infinite graphs. They proved that for every connected infinite graph we have D (G) ∆, where ∆ is a cardinal number such that degree of any vertex is at most ∆.…”

Breaking graph symmetries by edge colourings

“…While Tucker's conjecture is still wide open, there are many partial results towards it, see [3,6,7,10,11,14,16]. Broere and Pilśniak [2] noticed that most of these partial results can be generalised to edge colourings. Consequently, they conjecured that an analogous statement to Conjecture 1 should hold in the realm of edge colourings.…”

“…Conjecture 2 (Infinite edge motion conjecture [2]). Let G be a countable, connected graph and assume that every automorphism of G moves infinitely many edges.…”