Mycielskian

In the mathematical area of graph theory, the Mycielskian or Mycielski graph of an undirected graph is a larger graph formed from it by a construction of Jan Mycielski (1955). The construction preserves the property of being triangle-free but increases the chromatic number; by applying the construction repeatedly to a triangle-free starting graph, Mycielski showed that there exist triangle-free graphs with arbitrarily large chromatic number.

Let the n vertices of the given graph G be v₀, v₁, etc. The Mycielski graph μ(G) of G contains G itself as an isomorphic subgraph, together with n+1 additional vertices: a vertex u_i corresponding to each vertex v_i of G, and another vertex w. Each vertex u_i is connected by an edge to w, so that these vertices form a subgraph in the form of a star K_1,n. In addition, for each edge v_iv_j of G, the Mycielski graph includes two edges, u_iv_j and v_iu_j.

Thus, if G has n vertices and m edges, μ(G) has 2n+1 vertices and 3m+n edges.

The illustration shows Mycielski's construction as applied to a 5-vertex cycle graph with vertices v_i for 0 ≤ i ≤ 4. The resulting Mycielskian is the Grötzsch graph, an 11-vertex graph with 20 edges. The Grötzsch graph is the smallest triangle-free 4-chromatic graph (Chvátal 1974).

Applying the Mycielskian repeatedly, starting with a graph with a single edge, produces a sequence of graphs M_i = μ(M_i-1), also sometimes called the Mycielski graphs. The first few graphs in this sequence are the graph M₂ = K₂ with two vertices connected by an edge, the cycle graph M₃ = C₅, and the Grötzsch graph with 11 vertices and 20 edges.

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