---
tag: 1YZ7NV161A
kind: concept
name: Perfect graph
msc: [05C17]
sources:
  - url: https://en.wikipedia.org/wiki/Perfect_graph
same_as:
  wikidata: Q1187620
  wikipedia: Perfect graph
---

## Summary

A perfect graph is a graph in which, for every induced subgraph, the [[chromatic number|0VHJRXNAAS]] equals the size of the largest [[clique|0BYX1D7C2E]].
For every graph the chromatic number is at least the [[clique number|04MQSQ1F5Q]], and perfect graphs are those where this lower bound is always exact.
Bipartite graphs, interval graphs and comparability graphs are perfect.
The [[strong perfect graph theorem|MFNNZQ484T]] characterizes perfect graphs by forbidden induced subgraphs.
On perfect graphs, colouring and finding a largest clique can be done in polynomial time.

## Explanations

### L1 human

To colour a graph you need at least as many colours as the largest group of mutually adjacent vertices.
In a perfect graph this is always enough, not just for the whole graph but for every piece of it you get by deleting vertices.
The $5$-cycle is the smallest graph that is not perfect: it has no triangle, so its largest clique has size $2$, but it needs $3$ colours.