Coloring graphs as complete graph invariants

Abstract

We investigate the extent to which the k-coloring graph π’žπ‘˜β‘(𝐺) uniquely determines the base graph G and the number of colors k. The vertices of π’žπ‘˜β‘(𝐺) are the proper k-colorings of G, and edges connect colorings that differ on exactly one vertex. There are nonisomorphic graphs 𝐺1 and 𝐺2 with isomorphic coloring graphs, so π’žπ‘˜β‘(𝐺) is not a complete invariant in general. However, for color palettes with surplus colors (when the number of colors k is greater than the chromatic number), we prove that the coloring graph is a complete invariant. Specifically, provided that π‘˜1> πœ’β‘(𝐺1), we show that π’žπ‘˜1⁑(𝐺1) β‰…π’žπ‘˜2⁑(𝐺2) implies 𝐺1 ≅𝐺2 and π‘˜1= π‘˜2. Thus, there is a natural bijection between pairs (𝐺,π‘˜) with π‘˜> πœ’β‘(𝐺) and their coloring graphs π’žπ‘˜β‘(𝐺). Furthermore, no coloring graph of the form π’žπœ’β‘(𝐺)⁑(𝐺) is isomorphic to a coloring graph with surplus colors. Our constructive proof provides a method to decide whether a coloring graph is generated with surplus colors, although the resulting algorithms are inefficient.

Publisher

Elsevier

Publication Date

11-2026

Publication Title

Discrete Mathematics

Department

Computer Science

Document Type

Article

DOI

https://doi.org/10.1016/j.disc.2026.115245

Keywords

Coloring graphs, Reconfiguration systems, Graph invariants

Language

English

Format

text

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