Independent Coverings and Orthogonal Colourings

TL;DR

This paper disproves two open conjectures related to independent coverings of sparse graphs and orthogonal colourings of trees, establishing a new relation and providing conditions for their existence.

Contribution

It introduces a relation between independent coverings and orthogonal colourings and provides new degree conditions for independent coverings in certain sparse graphs.

Findings

Disproved two open conjectures in graph theory.

Established a relation between independent coverings and orthogonal colourings.

Identified degree conditions for independent coverings in graphs with a square number of vertices.

Abstract

In this paper, two open conjectures are disproved. One conjecture regards independent coverings of sparse partite graphs, whereas the other conjecture regards orthogonal colourings of tree graphs. A relation between independent coverings and orthogonal colourings is established. This relation is applied to find independent coverings of some sparse partite graphs. Additionally, a degree condition providing the existence of an independent covering in the case where the graph has a square number of vertices is found.