Bounds for Distinguishing Invariants of Infinite Graphs

Abstract

We consider infinite graphs. The distinguishing number D(G) of a graph G is the minimum number of colours in a vertex colouring of G that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by D (G). We prove that D (G) D(G) + 1. For proper colourings, we study relevant invariants called the distinguishing chromatic number χ D (G), and the distinguishing chromatic index χ D (G), for vertex and edge colourings, respectively. We show that χ D (G) 2∆(G) − 1 for graphs with a finite maximum degree ∆(G), and we obtain substantially lower bounds for some classes of graphs with infinite motion. We also show that χ D (G) χ (G) + 1, where χ (G) is the chromatic index of G, and we prove a similar result χ D (G) χ (G) + 1 for proper total colourings. A number of conjectures are formulated.

Distinguishing Graphs of Maximum Valence 3

“…In [15] the chromatic distinguishing number of infinite graphs is investigated. For connected graphs of bounded valence d it is shown that χ D (G) 2d − 1, and for infinite subcubic graphs of infinite motion this improves to χ D (G) 4.…”

The distinguishing number and distinguishing chromatic number for posets

“…We will define two analogs of a proper coloring, each of which leads to a distinguishing chromatic number. The analogous parameter for graphs, χ D (G), is introduced in [8] and studied further by other authors, see for example [3,4,7,10]. Definition 23.…”

On symmetries of edge and vertex colourings of graphs

“…The distinguishing index D (G) of a graph G is the smallest number of colours such that there is an edge colouring which is not preserved by any non-identity automorphism. Many results about distinguishing numbers hold for distinguishing indices as well, sometimes with almost identical proofs (see [3,8,9], for example). Furthermore, there are problems such as Tucker's Infinite Motion Conjecture [12] that are still wide open for vertex colourings, but whose edge colouring version has a relatively simple proof [10].…”

“…It was first proved for finite graphs in [9]. See [8] for an extension to infinite graphs and [10] for an alternative proof for both finite and infinite graphs. It was shown in [8] that Theorem 1.1 remains true even if D(G) is an arbitrary infinite cardinal.…”

“…See [8] for an extension to infinite graphs and [10] for an alternative proof for both finite and infinite graphs. It was shown in [8] that Theorem 1.1 remains true even if D(G) is an arbitrary infinite cardinal. Since α = α + 1 for any infinite cardinal α, this implies that for any graph G with infinite distinguishing number we have D (G) ≤ D(G).…”