Extremal graphs for blow-ups of cycles and trees

Abstract: The \emph{blow-up} of a graph $H$ is the graph obtained from replacing each edge in $H$ by a clique of the same size where the new vertices of the cliques are all different. Erd\H{o}s et al. and Chen et al. determined the extremal number of blow-ups of stars. Glebov determined the extremal number and found all extremal graphs for blow-ups of paths. We determined the extremal number and found the extremal graphs for the blow-ups of cycles and a large class of trees, when $n$ is sufficiently large. This generalizes their results. The additional aim of our note is to draw attention to a powerful tool, a classical decomposition theorem of Simonovits.