Ban–Linial's Conjecture and treelike snarks

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