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.
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