The Tur\'an number for the edge blow-up of trees: the missing case

TL;DR

This paper completes the determination of the Turán number for the edge blow-up of trees by solving the last remaining case, advancing understanding in extremal graph theory.

Contribution

It provides the exact Turán number for the previously unresolved case of the edge blow-up of trees, answering an open problem in the field.

Findings

Determined the Turán number for the missing case of edge blow-up of trees.

Extended the classification of extremal graphs for edge blow-ups.

Resolved an open problem posed by Wang, Hou, Liu, and Ma.

Abstract

The edge blow-up of a graph is the graph obtained from replacing each edge of it by a clique of the same size where the new vertices of the cliques are all different. Wang, Hou, Liu and Ma determined the Tur\'{a}n number of the edge blow-up of trees except one particular case. Answering an problem posed by them, we determined the Tur\'{a}n number of this particular case.