Mycielskian

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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[edit]

Mycielskian construction applied to a 5-cycle graph, producing the Grötzsch graph with 11 vertices and 20 edges, the smallest triangle-free 4-chromatic graph (Chvátal 1974).

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 , 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[edit]

M2, M3 and M4 Mycielski graphs

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).

Properties[edit]

Hamiltonian cycle in M4 (Grötzsch graph)

Cones over graphs[edit]

A generalized Mycielskian, formed as a cone over the 5-cycle, Δ3(C5) = Δ32(K2)).

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.

References[edit]

  • Chvátal, Vašek (1974), "The minimality of the Mycielski graph", Graphs and Combinatorics (Proc. Capital Conf., George Washington Univ., Washington, D.C., 1973), Lecture Notes in Mathematics, 406, Springer-Verlag, pp. 243–246.
  • Došlić, Tomislav (2005), "Mycielskians and matchings", Discussiones Mathematicae Graph Theory, 25 (3): 261–266, doi:10.7151/dmgt.1279, MR 2232992.
  • Fisher, David C.; McKenna, Patricia A.; Boyer, Elizabeth D. (1998), "Hamiltonicity, diameter, domination, packing, and biclique partitions of Mycielski's graphs", Discrete Applied Mathematics, 84 (1–3): 93–105, doi:10.1016/S0166-218X(97)00126-1.
  • Lin, Wensong; Wu, Jianzhuan; Lam, Peter Che Bor; Gu, Guohua (2006), "Several parameters of generalized Mycielskians", Discrete Applied Mathematics, 154 (8): 1173–1182, doi:10.1016/j.dam.2005.11.001.
  • Mycielski, Jan (1955), "Sur le coloriage des graphes" (PDF), Colloq. Math., 3: 161–162.
  • Stiebitz, M. (1985), Beiträge zur Theorie der färbungskritschen Graphen, Habilitation thesis, Technische Universität Ilmenau. As cited by Tardif (2001).
  • Tardif, C. (2001), "Fractional chromatic numbers of cones over graphs", Journal of Graph Theory, 38 (2): 87–94, doi:10.1002/jgt.1025.

External links[edit]