Extremal graphs for blow-ups of cycles and trees

TL;DR

This paper determines the maximum number of edges in large graphs avoiding blow-ups of cycles and certain trees, extending previous results and highlighting a classical decomposition theorem as a key tool.

Contribution

It generalizes extremal graph results to blow-ups of cycles and a broad class of trees, using Simonovits' decomposition theorem.

Findings

Exact extremal numbers for blow-ups of cycles and trees.

Characterization of extremal graphs for these structures.

Application of Simonovits' decomposition theorem as a powerful method.

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.