# Errera graph

Errera graph
The Errera graph
Named after Alfred Errera
Vertices 17
Edges 45
Diameter 4
Girth 3
Automorphisms 20 (D10)
Chromatic number 4
Chromatic index 6
Properties Planar
Hamiltonian[1]

In the mathematical field of graph theory, the Errera graph is a graph with 17 vertices and 45 edges discovered by Alfred Errera.[2] Published in 1921, it provides an example of how Kempe's proof of the four color theorem cannot work.[3][4]

Later, the Fritsch graph and Soifer graph provide two smaller counterexamples.[5]

The Errera graph is planar and has chromatic number 4, chromatic index 6, radius 3, diameter 4 and girth 3. All its vertices are of degree 5 or 6 and it is a 5-vertex-connected graph and a 5-edge-connected graph.

## Algebraic properties

The Errera graph is not a vertex-transitive graph and its full automorphism group is isomorphic to the dihedral group of order 20, the group of symmetries of a decagon, including both rotations and reflections.

The characteristic polynomial of the Errera graph is $-(x^2-2 x-5) (x^2+x-1)^2 (x^3-4 x^2-9 x+10) (x^4+2 x^3-7 x^2-18 x-9)^2$.

## References

1. ^
2. ^
3. ^ Errera, A. "Du coloriage des cartes et de quelques questions d'analysis situs." Ph.D. thesis. 1921.
4. ^ Peter Heinig. Proof that the Errera Graph is a narrow Kempe-Impasse. 2007.
5. ^ Gethner, E. and Springer, W. M. II. "How False Is Kempe's Proof of the Four-Color Theorem?" Congr. Numer. 164, 159-175, 2003.