What are Generalized Mycielskians (Cones over graphs) - Graph Theory Tutorial
This video introduces Generalized Mycielskians with visual examples. We will look at the generalized Mycielskian from two perspectives: the "cone over a graph," or "cone construction" as defined by Tardif in 2001 using graph products, and the generalized Mycielskian outlined by Lin et. al in 2006 using only set operations (with slight variations in notation for pedagogical purposes). The latter generalizes the Mycielskian for graphs with isolated vertices and is equivalent to the former when the graphs do not have isolated vertices.
The generalized Mycielskian, or m-Mycielskian of a graph G without isolated vertices is the tensor product of a path graph of length m with graph G. The m-Mycielskians can be used to construct graphs of arbitrarily high chromatic number and odd girth, much like the Mycielskian (the m = 2 case of the m-Mycielskians) can be used to construct graphs of arbitrarily high chromatic number.
For more information, see these links:
https://en.wikipedia.org/wiki/Myciels...
https://www.sciencedirect.com/science...
https://onlinelibrary.wiley.com/doi/a...
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