  The family of snarks -- connected bridgeless cubic graphs that cannot be
3-edge-coloured -- is well-known as a potential source of counterexamples to
several important and long-standing conjectures in graph theory. These include
the cycle double cover conjecture, Tutte's 5-flow conjecture, Fulkerson's
conjecture, and several others. One way of approaching these conjectures is
through the study of structural properties of snarks and construction of small
examples with given properties. In this paper we deal with the problem of
determining the smallest order of a nontrivial snark (that is, one which is
cyclically 4-edge-connected and has girth at least 5) of oddness at least 4.
Using a combination of structural analysis with extensive computations we prove
that the smallest order of a snark with oddness at least 4 and cyclic
connectivity 4 is 44. Formerly it was known that such a snark must have at
least 38 vertices [J. Combin. Theory Ser. B 103 (2013), 468--488] and one such
snark on 44 vertices was constructed by Lukot'ka et al. [Electron. J. Combin.
22 (2015), #P1.51]. The proof requires determining all cyclically
4-edge-connected snarks on 36 vertices, which extends the previously compiled
list of all such snarks up to 34 vertices [J. Combin. Theory Ser. B, loc.
cit.]. As a by-product, we use this new list to test the validity of several
conjectures where snarks can be smallest counterexamples.
