The Errera graph is the 17-node planar graph illustrated above that tangles the Kempe chains in Kempe's coloring algorithm and thus provides an example of how Kempe's (1879) supposed proof of the four-color theorem fails.
Gethner et al. (2009) successfully found four-color colorings of the Errera graph on all tested runs of their randomized Kempe-Kittell coloring algorithm. The largest observed number of randomly selected Kempe-Kittell switches needed to resolve a Kempe impasse at a single vertex was 73. This is an experimental maximum, not a proved universal upper bound. Their 100-switch cutoff was an implementation limit. These results concern a repair of Kempe's coloring algorithm, rather than its original faulty argument.
Xie and Bowling (2026) classify the Kempe impasse colorings of the Errera planar map with a central pentagonal region left uncolored and give explicit repairs using Kittell's Kempe chain exchanges. They also extend the repair to specified variations of the planar map. This Errera-specific resolution does not establish that the Kempe-Kittell coloring algorithm always succeeds on an arbitrary planar graph.
The Errera graph is implemented in the Wolfram Language as GraphData["ErreraGraph"].
The Fritsch graph and Soifer graph provide smaller (and in fact the smallest possible) counterexamples.
A number of other drawings (many of which are vertex-vertex and/or edge-vertex degenerate) are illustrated above.
The Errera graph has no planar graph embedding that is a unit-distance embedding (since
it contains the 9-node triangular
cupola forbidden graph for unit-distance
embeddings), but a beautiful three-dimensional unit-distance
embedding can be obtained from two oppositely-oriented copies of a gyroelongated
pentagonal pyramid, i.e., a truncated regular
icosahedron with one vertex and adjoining
faces removed, adjoined at their pentagonal faces
(E. Weisstein, Mar. 8, 2022). This is related to its being the dual
graph of the 
-fullerene, which is one of the three fullerenes
on 30 vertices.
