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

TL;DR

This paper investigates the Turán problem for the edge blow-up of trees, extending previous results to all trees with certain degree conditions, and determines extremal graphs for specific forbidden subgraph families.

Contribution

It extends the Turán number results to all trees with minimum degree at least two in the smaller color class and characterizes extremal graphs for certain forbidden configurations.

Findings

Complete resolution of the Turán problem for edge blow-ups of trees except one case.

Determination of maximum edges and extremal graphs for certain forbidden subgraph families.

Extension of classical results to broader classes of trees and graphs.

Abstract

The edge blow-up of a graph $F$ is the graph obtained from replacing each edge in $F$ by a clique of the same size where the new vertices of the cliques are all different. In this article, we concern about the Tur\'an problem for the edge blow-up of trees. Erd\H{o}s et al. (1995) and Chen et al. (2003) solved the problem for stars. The problem for paths was resolved by Glebov (2011). Liu (2013) extended the above results to cycles and a special family of trees with the minimum degree at most two in the smaller color class (paths and proper subdivisions of stars were included in the family). In this article, we extend Liu's result to all the trees with the minimum degree at least two in the smaller color class. Combining with Liu's result, except one particular case, the Tur\'an problem for the edge blow-up of trees is completely resolved. Moreover, we determine the maximum number of edges in the family of $\{K_{1,k}, kK_2, 2K_{1,k-1}\}$-free graphs and the extremal graphs, which is an extension of a result given by Abbott et al. (1972).