Distinguishing graphs of maximum valence 3

TL;DR

This paper classifies all connected graphs with maximum degree 3 that require exactly 3 colors for a distinguishing coloring, and shows all infinite such graphs are 2-distinguishable.

Contribution

It provides a complete classification of connected graphs with maximum degree 3 and distinguishing number 3, including infinite graphs.

Findings

All infinite connected graphs with maximum degree 3 are 2-distinguishable.

Complete classification of connected graphs with maximum degree 3 and distinguishing number 3.

Implication for symmetry breaking in graphs with degree 3.

Abstract

The distinguishing number $D(G)$ of a graph $G$ is the smallest number of colors that is needed to color $G$ such that the only color preserving automorphism is the identity. We give a complete classification for all connected graphs $G$ of maximum valence $\triangle(G)=3$ and distinguishing number $D(G) = 3$. As one of the consequences we get that all infinite connected graphs with $\triangle(G)=3$ are 2-distinguishable.