mycielskian

mycielskian(G, iterations=1)

Returns the Mycielskian of a simple, undirected graph G

The Mycielskian of graph preserves a graph’s triangle free property while increasing the chromatic number by 1.

The Mycielski Operation on a graph, \(G=(V, E)\), constructs a new graph with \(2|V| + 1\) nodes and \(3|E| + |V|\) edges.

The construction is as follows:

Let \(V = {0, ..., n-1}\). Construct another vertex set \(U = {n, ..., 2n}\) and a vertex, w. Construct a new graph, M, with vertices \(U \bigcup V \bigcup w\). For edges, \((u, v) \in E\) add edges \((u, v), (u, v + n)\), and \((u + n, v)\) to M. Finally, for all vertices \(u \in U\), add edge \((u, w)\) to M.

The Mycielski Operation can be done multiple times by repeating the above process iteratively.

More information can be found at https://en.wikipedia.org/wiki/Mycielskian

Parameters

Ggraph

A simple, undirected NetworkX graph

iterationsint

The number of iterations of the Mycielski operation to perform on G. Defaults to 1. Must be a non-negative integer.

Returns

Mgraph

The Mycielskian of G after the specified number of iterations.

Notes

Graph, node, and edge data are not necessarily propagated to the new graph.