Colourings of cubic graphs inducing isomorphic monochromatic subgraphs
Marien Abreu, Jan Goedgebeur, Domenico Labbate, Giuseppe Mazzuoccolo

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 -bisection of a bridgeless cubic graph is a -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 . Ban and Linial conjectured that every bridgeless cubic graph admits a -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 with has a -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…
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Graph Labeling and Dimension Problems
