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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year

SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

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Volume 20, Issue 2


Simple Graphs as Simplicial Complexes: the Mycielskian of a Graph

Piotr Rudnicki / Lorna Stewart
Published Online: 2013-02-02 | DOI: https://doi.org/10.2478/v10037-012-0019-8


Harary [10, p. 7] claims that Veblen [20, p. 2] first suggested to formalize simple graphs using simplicial complexes. We have developed basic terminology for simple graphs as at most 1-dimensional complexes.

We formalize this new setting and then reprove Mycielski’s [12] construction resulting in a triangle-free graph with arbitrarily large chromatic number. A different formalization of similar material is in [15].

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

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

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

  • [4] Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563-567, 1990.Google Scholar

  • [5] Grzegorz Bancerek. Mizar analysis of algorithms: Preliminaries. Formalized Mathematics, 15(3):87-110, 2007, doi:10.2478/v10037-007-0011-x.CrossrefGoogle Scholar

  • [6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Google Scholar

  • [7] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Google Scholar

  • [8] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Google Scholar

  • [9] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Google Scholar

  • [10] Frank Harary. Graph theory. Addison-Wesley, 1969.Google Scholar

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

  • [12] J. Mycielski. Sur le coloriage des graphes. Colloquium Mathematicum, 3:161-162, 1955.Google Scholar

  • [13] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Google Scholar

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

  • [15] Piotr Rudnicki and Lorna Stewart. The Mycielskian of a graph. Formalized Mathematics, 19(1):27-34, 2011, doi: 10.2478/v10037-011-0005-6.CrossrefGoogle Scholar

  • [16] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.Google Scholar

  • [17] Andrzej Trybulec. Many sorted sets. Formalized Mathematics, 4(1):15-22, 1993.Google Scholar

  • [18] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1(1):187-190, 1990.Google Scholar

  • [19] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Google Scholar

  • [20] Oswald Veblen. Analysis Situs, volume V. AMS Colloquium Publications, 1931Google Scholar

About the article

This work has been partially supported by the NSERC grant OGP 9207

Published Online: 2013-02-02

Published in Print: 2012-12-01

Citation Information: Formalized Mathematics, Volume 20, Issue 2, Pages 161–174, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.2478/v10037-012-0019-8.

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