---
tag: HKJCK0GJHW
kind: concept
name: Uniquely colorable graph
aliases: [uniquely-edge-colorable-graph, uniquely-total-colorable-graph]
sources:
  - wikipedia: en:676304@r1352721310
    locator: Uniquely colorable graph
    note: adopted
    mention: wikipedia://en/676304
    sections: [Summary]
same_as:
  wikidata: Q7886960
  wikipedia: Uniquely colorable graph
inventory: [8H61EQJH7V]
---

## Summary

In graph theory, a **uniquely colorable graph** is a $k$-chromatic [[graph|00C0BR7D7H]] that has only one possible (proper) $k$-[[coloring|0NCPXH194Y]] up to [[permutation|05RWFKVTJ7]] of the colors. Equivalently, there is only one way to partition its vertices into $k$ [[independent sets|08ANQHHG3P]] and there is no way to partition them into $k - 1$ independent sets.