Bounds for Distinguishing Invariants of Infinite Graphs

TL;DR

This paper investigates invariants related to the symmetry-breaking vertex and edge colourings of infinite graphs, establishing bounds and relationships between these invariants and classical graph parameters.

Contribution

It introduces bounds for distinguishing invariants in infinite graphs, relating them to classical parameters and proposing new conjectures.

Findings

Proves that the distinguishing index D'(G) is at most D(G)+1.

Establishes upper bounds for the distinguishing chromatic number in terms of maximum degree.

Shows that the distinguishing chromatic index is at most one more than the chromatic index.

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 $D'(G)\leq D(G)+1$. For proper colourings, we study relevant invariants called the distinguishing chromatic number $\chi_D(G)$, and the distinguishing chromatic index $\chi'_D(G)$, for vertex and edge colourings, respectively. We show that $\chi_D(G)\leq 2\Delta(G)-1$ for graphs with a finite maximum degree $\Delta(G)$, and we obtain substantially lower bounds for some classes of graphs with infinite motion. We also show that $\chi'_D(G)\leq \chi'(G)+1$, where $\chi'(G)$ is the chromatic index of $G$, and we prove a similar result $\chi''_D(G)\leq \chi''(G)+1$ for proper total colourings. A number of conjectures are formulated.