# The Mycielskian of a Graph

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## The Mycielskian of a Graph

Let ω(G) and χ(G) be the clique number and the chromatic number of a graph G. Mycielski [11] presented a construction that for any n creates a graph Mn which is triangle-free (ω(G) = 2) with χ(G) > n. The starting point is the complete graph of two vertices (K2). M(n+1) is obtained from Mn through the operation μ(G) called the Mycielskian of a graph G.

We first define the operation μ(G) and then show that ω(μ(G)) = ω(G) and χ(μ(G)) = χ(G) + 1. This is done for arbitrary graph G, see also [10]. Then we define the sequence of graphs Mn each of exponential size in n and give their clique and chromatic numbers.

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.

[3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.

[4] Grzegorz Bancerek. Bounds in posets and relational substructures. Formalized Mathematics, 6(1):81-91, 1997.

[5] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.

[6] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.

[7] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.

[8] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.

[9] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.

[10] M. Larsen, J. Propp, and D. Ullman. The fractional chromatic number of Mycielski's graphs. Journal of Graph Theory, 19:411-416, 1995.

[11] J. Mycielski. Sur le coloriage des graphes. Colloquium Mathematicum, 3:161-162, 1955.

[12] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.

[13] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.

[14] Krzysztof Retel. The class of series - parallel graphs. Part I. Formalized Mathematics, 11(1):99-103, 2003.

[15] Piotr Rudnicki. Dilworth's decomposition theorem for posets. Formalized Mathematics, 17(4):223-232, 2009, doi: 10.2478/v10037-009-0028-4.

[16] Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski - Zorn lemma. Formalized Mathematics, 1(2):387-393, 1990.

[17] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.

[18] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.

[19] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

# Formalized Mathematics

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