A note on 2--bisections of claw--free cubic graphs

TL;DR

This paper proves Ban--Linial's conjecture that all bridgeless cubic graphs, except the Petersen graph, admit a 2-bisection, specifically confirming the conjecture for claw-free cubic graphs.

Contribution

The paper establishes the validity of Ban--Linial's conjecture for claw-free cubic graphs, a significant subclass of bridgeless cubic graphs.

Findings

Ban--Linial's conjecture holds for claw-free cubic graphs.

The Petersen graph is the only exception among bridgeless cubic graphs.

Claw-free property facilitates the 2-bisection construction.

Abstract

A \emph{$k$--bisection} of a bridgeless cubic graph $G$ is a $2$--colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes have order at most $k$. Ban and Linial conjectured that {\em every bridgeless cubic graph admits a $2$--bisection except for the Petersen graph}. In this note, we prove Ban--Linial's conjecture for claw--free cubic graphs.