Tur\'an Number of Subdivisions of Multipartite Graphs

TL;DR

This paper studies the maximum edge count in graphs avoiding certain subdivided multipartite graphs, providing new upper bounds for these extremal problems, especially for subdivisions of bipartite and tripartite graphs.

Contribution

It establishes new upper bounds on the Turán number for 1-subdivisions of specific multipartite graphs, extending extremal graph theory results.

Findings

Upper bound for Turán number of 1-subdivision of K_{s,t}^+

Upper bound for extremal number of subdivisions of certain tripartite graphs

Results apply to graphs with 4 ≤ s ≤ t

Abstract

In this paper, we investigate the Tur\'an exponent for $1$-subdivisions of graphs that are neither bipartite nor complete. Specifically, we establish an upper bound on the Tur\'an number of the 1-subdivision of $K_{s,t}^+$, where $K_{s,t}^+$ is obtained by adding a single edge within the part of size $s$ of the complete bipartite graph $K_{s,t}$, with $4\leq s \leq t$. In addition, we derive an upper bound for the extremal number of a family of graphs formed by (possibly degenerate) 1-subdivisions of certain tripartite graphs.