The Mycielskian of a Graph

Piotr Rudnicki; Lorna Stewart

Formalized Mathematics (2011)

  • Volume: 19, Issue: 1, page 27-34
  • ISSN: 1426-2630

Abstract

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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.

How to cite

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Piotr Rudnicki, and Lorna Stewart. "The Mycielskian of a Graph." Formalized Mathematics 19.1 (2011): 27-34. <http://eudml.org/doc/267007>.

@article{PiotrRudnicki2011,
abstract = {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.},
author = {Piotr Rudnicki, Lorna Stewart},
journal = {Formalized Mathematics},
keywords = {clique number; chromatic number; Mycielskian},
language = {eng},
number = {1},
pages = {27-34},
title = {The Mycielskian of a Graph},
url = {http://eudml.org/doc/267007},
volume = {19},
year = {2011},
}

TY - JOUR
AU - Piotr Rudnicki
AU - Lorna Stewart
TI - The Mycielskian of a Graph
JO - Formalized Mathematics
PY - 2011
VL - 19
IS - 1
SP - 27
EP - 34
AB - 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.
LA - eng
KW - clique number; chromatic number; Mycielskian
UR - http://eudml.org/doc/267007
ER -

References

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