# Simple Graphs as Simplicial Complexes: the Mycielskian of a Graph

Formalized Mathematics (2012)

- Volume: 20, Issue: 2, page 161-174
- ISSN: 1426-2630

## Access Full Article

top## Abstract

top## How to cite

topPiotr Rudnicki, and Lorna Stewart. "Simple Graphs as Simplicial Complexes: the Mycielskian of a Graph." Formalized Mathematics 20.2 (2012): 161-174. <http://eudml.org/doc/267612>.

@article{PiotrRudnicki2012,

abstract = {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].},

author = {Piotr Rudnicki, Lorna Stewart},

journal = {Formalized Mathematics},

keywords = {clique number; chromatic number; Mycielskian},

language = {eng},

number = {2},

pages = {161-174},

title = {Simple Graphs as Simplicial Complexes: the Mycielskian of a Graph},

url = {http://eudml.org/doc/267612},

volume = {20},

year = {2012},

}

TY - JOUR

AU - Piotr Rudnicki

AU - Lorna Stewart

TI - Simple Graphs as Simplicial Complexes: the Mycielskian of a Graph

JO - Formalized Mathematics

PY - 2012

VL - 20

IS - 2

SP - 161

EP - 174

AB - 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].

LA - eng

KW - clique number; chromatic number; Mycielskian

UR - http://eudml.org/doc/267612

ER -

## References

top- [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. Zbl06213858
- [3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
- [4] Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563-567, 1990.
- [5] Grzegorz Bancerek. Mizar analysis of algorithms: Preliminaries. Formalized Mathematics, 15(3):87-110, 2007, doi:10.2478/v10037-007-0011-x.[Crossref]
- [6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
- [7] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
- [8] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
- [9] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
- [10] Frank Harary. Graph theory. Addison-Wesley, 1969.
- [11] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.
- [12] J. Mycielski. Sur le coloriage des graphes. Colloquium Mathematicum, 3:161-162, 1955. Zbl0064.17805
- [13] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.
- [14] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.
- [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.[Crossref] Zbl1276.05046
- [16] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.
- [17] Andrzej Trybulec. Many sorted sets. Formalized Mathematics, 4(1):15-22, 1993.
- [18] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1(1):187-190, 1990.
- [19] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
- [20] Oswald Veblen. Analysis Situs, volume V. AMS Colloquium Publications, 1931 Zbl0001.40604

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.