# 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 v$_1$, v$_2$, . . . , v$_n$. The Mycielski graph μ(G) 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$_i$v$_j$ of G, the Mycielski graph includes two edges, u$_i$v$_j$ and v$_i$u$_j$.
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 v$_i$v$_j$u$_k$, where v$_i$v$_j$v$_k$ 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 $c(u_{i})\in \{1,2,\ldots ,k{-}1\}$ 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 M$_i$ = μ(M$_{i−1}$), sometimes called the Mycielski graphs. The first few graphs in this sequence are the graph M$_2$ = K$_2$ with two vertices connected by an edge, the cycle graph M$_3$ = C$_5$, and the Grötzsch graph M$_4$ with 11 vertices and 20 edges.
In general, the graph M$_i$ is triangle-free, (i−1)-vertex-connected, and i-chromatic. The number of vertices in M$_i$ for i ≥ 2 is 3 × 2$^{i−2}$ − 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).

## Properties

* If G has chromatic number k, then μ(G) has chromatic number k + 1 (Mycielski 1955).
* If G is triangle-free, then so is μ(G) (Mycielski 1955).
* More generally, if G has clique number ω(G), then μ(G) has clique number of the maximum among 2 and ω(G). (Mycielski 1955)
* If G is a factor-critical graph, then so is μ(G) (Došlić 2005). In particular, every graph M$_i$ for i ≥ 2 is factor-critical.
* If G has a Hamiltonian cycle, then so does μ(G) (Fisher, McKenna & Boyer 1998).
* If G has domination number γ(G), then μ(G) has domination number γ(G)+1 (Fisher, McKenna & Boyer 1998).

## 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 $\Delta _{i}(G)$ 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(G) = Δ$_2$(G).
While the cone construction does not always increase the chromatic number, Stiebitz (1985) proved that it does so when applied iteratively to K$_2$. That is, define a sequence of families of graphs, called generalized Mycielskians, as
    ℳ(2) = {K$_2$} and ℳ(k+1) = {$\Delta _{i}(G)$ | G ∈ ℳ(k), i ∈ $\mathbb {N}$}.
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 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.