Orthogonal Colourings of Tensor Graphs

TL;DR

This paper investigates perfect k-orthogonal colourings of tensor graphs, reformulates the problem as a tensor subgraph problem, and establishes conditions and bounds for such colourings.

Contribution

It introduces a new tensor subgraph formulation for perfect orthogonal colourings and provides bounds and conditions for tensor graphs to have these colourings.

Findings

Reformulation of the 2-orthogonal colouring problem as a tensor subgraph problem

Establishment that tensor graphs inherit perfect k-orthogonal colourings from component graphs

Provision of new conditions for tensor graphs to admit perfect k-orthogonal colourings

Abstract

In this paper, perfect k-orthogonal colourings of tensor graphs are studied. First, the problem of determining if a given graph has a perfect 2-orthogonal colouring is reformulated as a tensor subgraph problem. Then, it is shown that if two graphs have a perfect $k$-orthogonal colouring, then so does their tensor graph. This provides an upper bound on the $k$-orthogonal chromatic number for general tensor graphs. Lastly, two other conditions for a tensor graph to have a perfect $k$-orthogonal colouring are given.