Let 
be a planar
graph whose vertices have been properly colored and suppose 
is colored 
. Define the 
-Kempe chain containing 
to be the maximal connected component of 
that
1. Contains 
,
and
2. Contains only vertices that are colored with elements from 
(Gethner and Springer 2003).
A number of small graphs (with 
the vertex count) that tangle
the chains in Kempe's algorithm and so provide examples of how Kempe's supposed proof
of the four-color theorem fails are illustrated
above and summarized in the following table.
 | graph name |
| 9 | Fritsch graph |
| 9 | Soifer graph |
| 15 | Poussin graph |
| 17 | Errera graph |
| 23 | Kittell graph |
| 25 | Heawood four-color graph |
Interestingly, a number of these examples (though not the Soifer graph, which contains a 4-cycle) correspond to the skeletons of (possibly degenerate) deltahedra (E. Weisstein, Mar. 7, 2022). In particular, the Fritsch graph is the skeleton of the triaugmented triangular prism and the Errera graph is the skeleton of two pentagon-adjoined gyroelongated pentagonal pyramids.
