Flows and bisections in cubic graphs

The existence of a bisection of the vertex set of a cubic graph G with
small monochromatic components is strictly related to the existence of
certain flows. In particular, a circular nowhere-zero r-flow in G
implies a bisection, where every connected subgraph on r-1 vertices
intersects both parts of the bisection. This is related to a recent
conjecture of Ban and Linial, stating that any bridgeless cubic graph,
other than the Petersen graph, admits a bisection, where the graph
induced by each part of the bisection consists of connected components
on at most two vertices. Here, we present some recent progress on Ban
and Linial conjecture.