---
tag: 6DG5BBQPRA
kind: concept
name: Penny graph
msc: [05C62, 52C15, 52C10]
specializes: [0NC9R64TXP]
sources:
  - url: https://en.wikipedia.org/wiki/Penny_graph
same_as:
  wikidata: Q30592245
  wikipedia: Penny graph
---

## Summary

A penny graph is the contact graph of a set of equal coins lying flat on a table without overlapping.
Each coin is a vertex, and two coins are joined when they touch.
Penny graphs are [[planar graphs|0NC9R64TXP]] with maximum degree at most $6$, and they are a test case for questions about unit distances and [[circle packings|S360JBEYKD]].

## Explanations

### L1 human

Lay some identical coins on a table so that none overlap.
Draw a dot at the centre of each coin and a line between the centres of two coins that touch.
The drawing is a penny graph.
Six coins around a central coin, all touching it and their neighbours, give a hexagon with its centre, a penny graph with $7$ vertices and $12$ edges.
No coin can touch more than six others, since a seventh would not fit around it.