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

Abstract

The distinguishing index D′(G) of a graph G is the least cardinal number d such that G has an edge‐coloring with d colors, which is preserved only by the trivial automorphism. We prove a general upper bound D′(G)goodbreakinfix≤normalΔgoodbreakinfix−1 for any connected infinite graph G with finite maximum degree normalΔgoodbreakinfix≥3. This is in contrast with finite graphs since for every normalΔgoodbreakinfix≥3 there exist infinitely many connected, finite graphs G with D′(G)goodbreakinfix=normalΔ. We also give examples showing that this bound is sharp for any maximum degree normalΔ.

On Asymmetric Colourings of Claw-Free Graphs

“…Substituting a central vertex v 0 by a central edge e 0 one obtains a bisymmetric tree. For infinite graphs, Pilśniak and Stawiski [18] showed that D (G) ∆(G)−1 whenever G is connected and ∆(G) 3.…”

A bound for the distinguishing index of regular graphs

“…[6]) that there is such a colouring for complete graphs on at least 3 vertices. Furthermore, by results from [8] and [9], it is known that D ′ (G) ≤ max{∆(G) − 1, 3}, and the proofs show that if G is regular of degree ∆ ≤ 4, then there are distinguishing edge colourings satisfying ( * ). Therefore, we may assume that G is not complete, that ∆ ≥ 5, and any ∆ ′ -regular graphs with ∆ ′ < ∆ has a distinguishing edge colouring with 3 colours which satisfies ( * ).…”

“…For connected graphs with finite maximum degree ∆, it is known that D ′ (G) ≤ ∆ unless G is C 3 , C 4 or C 5 . This bound is sharp, and the graphs which attain it are fully characterised, see [8,9]. On the other hand, there are many graph classes where better bounds are possible.…”

A class of graphs with distinguishing index $\bf D' \leq 3$

“…This was done in analogy to the concept of the distinguishing number, referring to coloring vertices rather than edges, which has been introduced and considered much earlier [1,3].The very first result on D ′ (G), obtained together with introducing the distinguishing index in [8], was that for connected graphs with finite maximum degree ∆, D ′ (G) ≤ ∆, unless G is C 3 , C 4 or C 5 . This was improved in [11,13] by characterizing graphs for which the equality holds. There are many classes of graphs for which much better bounds are possible.…”

“…The very first result on D ′ (G), obtained together with introducing the distinguishing index in [8], was that for connected graphs with finite maximum degree ∆, D ′ (G) ≤ ∆, unless G is C 3 , C 4 or C 5 . This was improved in [11,13] by characterizing graphs for which the equality holds. There are many classes of graphs for which much better bounds are possible.…”