The Descartes snarks are a set of (strong) snarks on 210 vertices and 315 edges discovered by William Tutte in 1948 writing under the pseudonym Blanche Descartes (Descartes 1948; Holton and Sheehan 1993, pp. 93-97). The Descartes snark illustrated above is implemented in the Wolfram Language as GraphData["DescartesSnark1"].
Descartes snarks are obtained by replacing each vertex of the Petersen graph with a 9-cycle and each edge with a graph related to the Petersen graph. This procedure can be performed in a number of ways, leading to multiple distinct Descartes snarks.
After quotienting by the side-preserving graph automorphism of each edge-replacement graph, there are 18 attachment states for each of the 15 edges. Thus, before quotienting by the global symmetries of the Petersen graph, the number of side-reduced labeled attachment assignments is
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A complete orbit calculation using the Cauchy-Frobenius lemma (also known as Burnside's lemma) and the 120 graph
automorphisms of the Petersen graph shows that
the construction yields exactly 
graph isomorphism classes
of Descartes snarks, where
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The construction decomposition is intrinsic: the only 5-cycles are the two internal 5-cycles in each 
-graph dipole. Their vertex incidences recover the 15
dipoles, after which the remaining vertices induce the ten 9-cycles. Therefore 
is also the unrestricted graph
isomorphism class count.
Every resulting graph has girth 5 and cyclic edge connectivity 4. A cyclic edge cut of size 4
is obtained from any graph dipole together with its
two incident three-vertex blocks of 9-cycle vertices, while an exact minimization
over all attachment states shows that no nontrivial edge
cut has fewer than four edges. Together with the proper-superposition result
below, this proves that all 
graphs are strong snarks; none is
merely a weak snark, and no attachment state fails
to produce a snark.
Máčajová and Škoviera (2021) interpret this construction as a proper snark superposition of the Petersen graph: the 9-cycles are supervertices, while
the edge-replacement graphs are proper 
-graph dipoles obtained
from the Petersen graph by deleting two vertices
at distance 2. More generally, every connected proper snark
superposition is itself a snark.
The Descartes snark is a 
cyclic group graph.
