Asymmetric $2$-colorings of graphs

TL;DR

This paper demonstrates that for most 3-connected planar graphs, edges can be 2-colored to eliminate automorphisms, and characterizes graphs with this property across all embeddings in orientable surfaces.

Contribution

It introduces a method for 2-coloring edges to break automorphisms in 3-connected planar graphs and characterizes all graphs with this property across all orientable surface embeddings.

Findings

All 3-connected planar graphs except K4 can be 2-colored to eliminate automorphisms.

Characterization of graphs with universal 2-coloring property across all embeddings.

The property holds regardless of the embedding surface.

Abstract

We show that the edges of every 3-connected planar graph except $K_4$ can be colored with two colors in such a way that the graph has no color preserving automorphisms. Also, we characterize all graphs which have the property that their edges can be $2$-colored so that no matter how the graph is embedded in any orientable surface, there is no homeomorphism of the surface which induces a non-trivial color preserving automorphism of the graph.