The Distinguishing Index of Infinite Graphs

Izak Broere,Monika Pilśniak

doi:10.37236/3933

Izak Broere, Monika Pilśniak

Open Access

https://doi.org/10.37236/3933

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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.

Highlights

Albertson and Collins [1] introduced thedistinguishing number D(G) of a graph G as the least cardinal d such that G has a labeling with d labels that is only preserved by the trivial automorphism
Given a graph G with an edge colouring f, we say that a graph automorphism φ : V (G) → V (G) of G preserves the edge colouring f if f = f (φ(x)φ(y)) for every edge xy ∈ E(G)
The distinguishing index D (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

Summary

Introduction

Albertson and Collins [1] introduced the (vertex-)distinguishing number D(G) of a graph G as the least cardinal d such that G has a labeling with d labels that is only preserved by the trivial automorphism This concept has spawned numerous papers, mostly on finite graphs. The distinguishing index D (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. Since G is connected, one can recover the original vertex colouring of the graph up to a permutation of the colours by knowing the colour of v It follows that any automorphism preserving edge colours either interchanges colours, and no such automorphism exists, or preserves the colours of the vertices, and is the identity. We obtain general theorems for countable and uncountable graphs, and a relationship with the Infinite Motion Lemma

A general bound for infinite graphs

Trees and tree-like graphs

The countable random graph

The Edge-Motion Lemma