Graph Classes Generated by Mycielskians

In this paper we use the classical notion of weak Mycielskian $ M^′ (G) $ of a graph $G$ and the following sequence: $M_0^' (G) = G$, $M_1^′ (G) = M^′ (G)$, and $M_n^' (G) = M^′ (M_n^′ − 1(G)) $, to show that if $G$ is a complete graph of order $p$, then the above sequence is a generator of the class of $p$-colorable graphs. Similarly, using Mycielskian $M(G)$ we show that analogously defined sequence is a generator of the class consisting of graphs for which the chromatic number of the subgraph induced by all vertices that belong to at least one triangle is at most $p$. We also address the problem of characterizing the latter class in terms of forbidden graphs.