The classic join of two graphs $G$ and $H$ results in $G \vee H$ whose chromatic number
$\chi(G \vee H)$ = $\chi(G)$ + $\chi(H)$.
$G \vee H$ = $G \cup H \cup$ Complete bipartite between vertices of $G$ and $H$.
This introduces $n_G.n_H$ new edges. I am looking for other operations which increases the chromatic number with a limited increase in the number of edges. The Mycielskian is one of them. Are there any other such simple tricks for increasing the chromatic number? I am looking for a construction where fewer triangles(new) would be formed. Any references on such works are welcomed.
Thank you for going through this.