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(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_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 only new triangles in μ(G) 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(G).

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 (k−1)-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(M_i−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 M_i is triangle-free, (i−1)-vertex-connected, and i-chromatic. The number of vertices in M_i for i ≥ 2 is 3 ÃƒÂ— 2^i−2 − 1 , while the number of edges for i ≥ 2 is , which begins as:

1, 5, 20, 71, 236, 755, 2360, 7271, 22196, 67355, ... .

Properties

• If G has chromatic number k, then μ(G) has chromatic number k + 1 .
• If G is triangle-free, then so is μ(G) .
• More generally, if G has clique number ω(G), then μ(G) has clique number of the maximum among 2 and ω(G).
• If G is a factor-critical graph, then so is μ(G) . In particular, every graph M_i for i Ã¢Â‰Â¥ 2 is factor-critical.
• If G has a Hamiltonian cycle, then so does μ(G) .
• If G has domination number γ(G), then μ(G) has domination number γ(G)+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 ÃƒÂ— 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, 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) = { | G ∈ ℳ(k), i ∈ }.

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.

References

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External links