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. 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.

Construction

Let the n vertices of the given graph G be v₁, v₂,..., v_n. The Mycielski graph μ contains G itself as a subgraph, together with n+1 additional vertices: a vertex u_i corresponding to each vertex v_i of G, and an extra 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 has 2n+1 vertices and 3m+''n edges.
The only new triangles in μ are of the form v''_iv_ju_k, where v_iv_jv_k is a triangle in G. Thus, if G is triangle-free, so is μ.
To see that the construction increases the chromatic number, consider a proper k-coloring of ; that is, a mapping with for adjacent vertices x,''y. If we had for all i'', then we could define a proper -coloring of G by when, and otherwise. But this is impossible for, so c must use all k colors for, and any proper coloring of the last vertex w must use an extra color. That is,.

Iterated Mycielskians

Applying the Mycielskian repeatedly, starting with the one-edge graph, produces a sequence of graphs M_i = μ, 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 M₄ with 11 vertices and 20 edges.
In general, the graph M_i is triangle-free, -vertex-connected, and i-chromatic. The number of vertices in M_i for i ≥ 2 is 3 × 2ⁱ⁻² − 1, while the number of edges for i ≥ 2 is, which begins as:

Properties

If G has chromatic number k, then μ has chromatic number k + 1.
If G is triangle-free, then so is μ.
More generally, if G has clique number ω, then μ has clique number of the maximum among 2 and ω.
If G is a factor-critical graph, then so is μ. In particular, every graph M_i for i ≥ 2 is factor-critical.
If G has a Hamiltonian cycle, then so does μ.
If G has domination number γ, then μ has domination number γ+1.

Cones over graphs

A generalization of the Mycielskian, called a cone over a graph, was introduced by and further studied by and. In this construction, one forms a graph from a given graph G by taking the tensor product G × H, where H is a path of length i with a self-loop at one end, and then collapsing into a single supervertex all of the vertices associated with the vertex of H at the non-loop end of the path. The Mycielskian itself can be formed in this way as μ = Δ₂.
While the cone construction does not always increase the chromatic number, proved that it does so when applied iteratively to K₂. That is, define a sequence of families of graphs, called generalized Mycielskians, as
For example, ℳ is the family of odd cycles. Then each graph in ℳ is k-chromatic. The proof uses methods of topological combinatorics developed by László Lovász to compute the chromatic number of Kneser graphs.
The triangle-free property is then strengthened as follows: if one only applies the cone construction Δ_i for i ≥ r, then the resulting graph has odd girth at least 2r + 1, that is, it contains no odd cycles of length less than 2r + 1.
Thus generalized Mycielskians provide a simple construction of graphs with high chromatic number and high odd girth.