Orthogonal Colourings of Cayley Graphs
Kyle MacKeigan, Jeannette Janssen
TL;DR
This paper investigates orthogonal colourings in Cayley graphs, determining orthogonal chromatic numbers for cycles, circulant graphs, and exploring product and Hamming graphs, advancing understanding of graph colourings.
Contribution
It provides the first complete determination of orthogonal chromatic numbers for cycle graphs and extends the study to circulant, product, and Hamming graphs.
Findings
Orthogonal chromatic number of cycle graphs is fully characterized.
Orthogonal chromatic numbers for certain circulant graphs are established.
Orthogonal colourings of product and Hamming graphs are analyzed.
Abstract
Two colourings of a graph are orthogonal if they have the property that when two vertices are coloured with the same colour in one colouring, then those vertices receive distinct colours in the other colouring. In this paper, orthogonal colourings of Cayley graphs are discussed. Firstly, the orthogonal chromatic number of cycle graphs are completely determined. Secondly, the orthogonal chromatic number of certain circulant graphs is explored. Lastly, orthogonal colourings of product graphs and Hamming graphs are studied.
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