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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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1898-9934
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The Mycielskian of a Graph

Piotr Rudnicki
  • University of Alberta, Edmonton, Canada
/ Lorna Stewart
  • University of Alberta, Edmonton, Canada
Published Online: 2011-07-18 | DOI: https://doi.org/10.2478/v10037-011-0005-6

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.

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  • [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.

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  • [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.

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  • [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. [Crossref]

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


Published Online: 2011-07-18

Published in Print: 2011-01-01


Citation Information: Formalized Mathematics. Volume 19, Issue 1, Pages 27–34, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.2478/v10037-011-0005-6, July 2011

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