Extremal graphs for blow-ups of cycles and trees

Abstract

The 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. Erdo{double acute}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 determine the extremal number and find the extremal graphs for the blow-ups of cycles and a large class of trees, when n is suffi ciently 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.