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.
@article{bwmeta1.element.doi-10_2478_v10037-011-0005-6,
author = {Piotr Rudnicki and Lorna Stewart},
title = {The Mycielskian of a Graph},
journal = {Formalized Mathematics},
volume = {19},
year = {2011},
pages = {27-34},
zbl = {1276.05046},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10037-011-0005-6}
}
Piotr Rudnicki; Lorna Stewart. The Mycielskian of a Graph. Formalized Mathematics, Tome 19 (2011) pp. 27-34. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10037-011-0005-6/
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