A Halin snark is a snark constructed from a cubic Halin graph by replacing the vertices of its perimeter cycle with auxiliary graph multipoles. The graph multipoles used by Máčajová and Škoviera (2021) are derived from a cubic graph with perfect matching cover index at least 5 and a bipartite cubic graph.
Every Halin snark has perfect matching cover index at least 5. When the fragments are constructed from cubic graphs of cyclic edge connectivity at least 4 and girth at least 5, the resulting Halin snarks are nontrivial snarks, i.e., cubic graphs of cyclic edge connectivity at least 4 and girth at least 5.
The family of Halin snarks contains the previously known nontrivial snarks with perfect matching cover index at
least 5, including treelike snarks, windmill snarks,
and the family constructed by Chen (Máčajová and Škoviera
2021). In particular, for every even number 
, there exists a nontrivial snark
on 
vertices
with perfect matching cover index at
least 5. The existence of nontrivial snarks of graph
orders 38 and 40 with perfect matching
cover index at least 5 remains open.
The 34- and 46-vertex perfect matching snarks, illustrated above, are Halin snarks that cannot be covered with four perfect matchings and have perfect matching cover index 5. They will be implemented in a future version of the Wolfram Language as GraphData["PerfectMatchingSnark34"] and GraphData["PerfectMatchingSnark46"].
The 46-vertex perfect matching snark has graph crossing number 12.
