Summary

Very hastily drawn image of the fourth Mycielski graph. Mycielski proved in 1955 that for every ${\displaystyle k\geq 1}$ there is a graph with chromatic number k that contains no triangle subgraphs, that is, whose maximal clique size is just 2. (This was also proved by A. A. Zykov, Mat. Zb. 24, (1949), 2, 163-183 (in Russian) and by "Blanche Descartes" (W. T. Tutte) Amer. Math. Monthly 61, (1954) p. 352.) The chromatic number of this graph is 4.

A simple construction of triangle-free graphs of any chromatic number is this. Start with a ${\displaystyle k}$-chromatic graph G with no triangle. Take ${\displaystyle k}$ copies of G. Form all sets consisting of 1 node from each copy. Connect the nodes in each of these sets to a new node. The resulting graph is triangle-free and ${\displaystyle k+1}$-chromatic.

If you can make the image look better, by all means do so.

Licensing

This work has been released into the public domain by its author, Bkell. This applies worldwide. In some countries this may not be legally possible; if so: Bkell grants anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

Original upload log