Simple Graphs as Simplicial Complexes: the Mycielskian of a Graph

Piotr Rudnicki; Lorna Stewart

Formalized Mathematics (2012)

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

Abstract

top
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].

How to cite

top

Piotr 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. [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990. 
  2. [2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. Zbl06213858
  3. [3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990. 
  4. [4] Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563-567, 1990. 
  5. [5] Grzegorz Bancerek. Mizar analysis of algorithms: Preliminaries. Formalized Mathematics, 15(3):87-110, 2007, doi:10.2478/v10037-007-0011-x.[Crossref] 
  6. [6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990. 
  7. [7] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990. 
  8. [8] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990. 
  9. [9] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990. 
  10. [10] Frank Harary. Graph theory. Addison-Wesley, 1969. 
  11. [11] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990. 
  12. [12] J. Mycielski. Sur le coloriage des graphes. Colloquium Mathematicum, 3:161-162, 1955. Zbl0064.17805
  13. [13] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990. 
  14. [14] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990. 
  15. [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. [16] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990. 
  17. [17] Andrzej Trybulec. Many sorted sets. Formalized Mathematics, 4(1):15-22, 1993. 
  18. [18] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1(1):187-190, 1990. 
  19. [19] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990. 
  20. [20] Oswald Veblen. Analysis Situs, volume V. AMS Colloquium Publications, 1931 Zbl0001.40604

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.