  A bridgeless cubic graph $G$ is said to have a 2-bisection if there exists a
2-vertex-colouring of $G$ (not necessarily proper) such that: (i) the colour
classes have the same cardinality, and (ii) the monochromatic components are
either an isolated vertex or an edge. In 2016, Ban and Linial conjectured that
every bridgeless cubic graph, apart from the well-known Petersen graph, admits
a 2-bisection. In the same paper it was shown that every Class I bridgeless
cubic graph admits such a bisection. The Class II bridgeless cubic graphs which
are critical to many conjectures in graph theory are known as snarks, in
particular, those with excessive index at least 5, that is, whose edge set
cannot be covered by four perfect matchings. Moreover, in [J. Graph Theory,
86(2) (2017), 149--158], Esperet et al. state that a possible counterexample to
Ban--Linial's Conjecture must have circular flow number at least 5. The same
authors also state that although empirical evidence shows that several graphs
obtained from the Petersen graph admit a 2-bisection, they can offer nothing in
the direction of a general proof. Despite some sporadic computational results,
until now, no general result about snarks having excessive index and circular
flow number both at least 5 has been proven. In this work we show that treelike
snarks, which are an infinite family of snarks heavily depending on the
Petersen graph and with both their circular flow number and excessive index at
least 5, admit a 2-bisection.
