A 2-Bisection with Small Number of Monochromatic Edges of a Claw-Free Cubic Graph

A bisection of a graph G is a partition of its vertex set into two parts of the same cardinality. A k -bisection of G is a bisection of G such that every component of each part has at most k vertices. Cui and Liu proved that every claw-free cubic graph contains a 2-bisection. In this paper, we improve this result showing that every claw-free cubic graph contains a 2-bisection with bounded number of monochromatic edges, where a monochromatic edge of a 2-bisection is an edge connecting two vertices of the same part of the 2-bisection. We also prove that our bound is best possible for all claw-free cubic simple graphs.