Colourings of cubic graphs inducing isomorphic monochromatic subgraphs

TL;DR

This paper investigates conjectures related to vertex and edge colourings of cubic graphs that produce isomorphic monochromatic subgraphs, providing theoretical and computational evidence, and proving the conjecture for specific graph classes.

Contribution

It offers new insights into longstanding conjectures, proves Ban-Linial's conjecture for cubic cycle permutation graphs, and explores the relationship between different colouring problems.

Findings

Evidence supporting Ban-Linial and Wormald conjectures.

Proved Ban-Linial's conjecture for cubic cycle permutation graphs.

Provided a negative answer to a problem on decompositions into linear forests.

Abstract

A $k$-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 (monochromatic components in what follows) have order at most $k$. Ban and Linial conjectured that every bridgeless cubic graph admits a $2$-bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph $G$ with $|E(G)| \equiv 0 \pmod 2$ has a $2$-edge colouring such that the two monochromatic subgraphs are isomorphic linear forests (i.e. a forest whose components are paths). Finally, Ando conjectured that every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic. In this paper, we give a detailed insight into the conjectures of Ban-Linial and Wormald and provide evidence of a strong relation of both of them with Ando's conjecture. Furthermore, we also give computational and theoretical evidence in their support. As a result, we pose some open problems stronger than the above mentioned conjectures. Moreover, we prove Ban-Linial's conjecture for cubic cycle permutation graphs. As a by-product of studying $2$-edge colourings of cubic graphs having linear forests as monochromatic components, we also give a negative answer to a problem posed by Jackson and Wormald about certain decompositions of cubic graphs into linear forests.