This thesis examines the issue of Fano colouring in cubic graphs, a field that establishes a strong link among edge-colouring theory, flow theory, and Steiner triple systems. We start by introducing the essential principles of graph theory pertinent to cubic graphs, such as perfect matchings, bridgeless graphs, and snarks, and then we review classical findings on Tait colourings. The Fano plane is presented as the smallest non-trivial Steiner triple system, and its structural characteristics are analyzed to establish the definition of Fano colourings.
The main section of the thesis centers on recognized findings is related to Fano coloring in bridgeless cubic graphs. We examine upper limits on the smallest quantity of lines in the Fano plane necessary for these colourings and emphasize significant unresolved conjectures, such as the Four-Line Conjecture and Fulkerson’s conjecture regarding six perfect matchings. The theoretical explanation is enhanced with illustrative instances, particularly the Petersen graph.
In the conclusion, we assess the existing theoretical framework and suggest potential avenues for future research, including computational investigations of Fano colourings for larger or more intricate snarks.
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