# 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.

## Construction

Let the n vertices of the given graph G be v1, v2, . . . , vn. The Mycielski graph μ(G) contains G itself as a subgraph, together with n+1 additional vertices: a vertex ui corresponding to each vertex vi of G, and an extra vertex w. Each vertex ui is connected by an edge to w, so that these vertices form a subgraph in the form of a star K1,n. In addition, for each edge vivj of G, the Mycielski graph includes two edges, uivj and viuj.

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

The only new triangles in μ(G) are of the form vivjuk, where vivjvk is a triangle in G. Thus, if G is triangle-free, so is μ(G).

To see that the construction increases the chromatic number $\chi (G)=k$ , consider a proper k-coloring of $\mu (G){-}\{w\}$ ; that is, a mapping $c:\{v_{1},\ldots ,v_{n},u_{1},\ldots ,u_{n}\}\to \{1,2,\ldots ,k\}$ with $c(x)\neq c(y)$ for adjacent vertices x,y. If we had $c(u_{i})\subset \{1,2,\ldots ,k{-}1\}$ for all i, then we could define a proper (k−1)-coloring of G by $c'\!(v_{i})=c(u_{i})$ when $c(v_{i})=k$ , and $c'\!(v_{i})=c(v_{i})$ otherwise. But this is impossible for $\chi (G)=k$ , so c must use all k colors for $\{u_{1},\ldots ,u_{n}\}$ , and any proper coloring of the last vertex w must use an extra color. That is, $\chi (\mu (G))=k{+}1$ .

## Iterated Mycielskians

Applying the Mycielskian repeatedly, starting with the one-edge graph, produces a sequence of graphs Mi = μ(Mi−1), 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 Mi is triangle-free, (i−1)-vertex-connected, and i-chromatic. The number of vertices in Mi for i ≥ 2 is 3 × 2i−2 − 1 (sequence A083329 in the OEIS), while the number of edges for i = 2, 3, . . . is:

1, 5, 20, 71, 236, 755, 2360, 7271, 22196, 67355, ... (sequence A122695 in the OEIS).

## Cones over graphs

A generalization of the Mycielskian, called a cone over a graph, was introduced by Stiebitz (1985) and further studied by Tardif (2001) and Lin et al. (2006). In this construction, one forms a graph Δi(G) from a given graph G by taking the tensor product of graphs 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(G) = Δ2(G).

While the cone construction does not always increase the chromatic number, Stiebitz (1985) proved that it does so when applied iteratively to K2. That is, define a sequence of families of graphs, called generalized Mycielskians, as

ℳ(2) = {K2} and ℳ(k+1) = {Δi(G) | G ∈ ℳ(k), i ∈ ℕ}.

For example, ℳ(3) is the family of odd cycles. Then each graph in ℳ(k) 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 Δi for ir, 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.