Turán Number of Subdivisions of Multipartite Graphs

Abstract

In this paper, we investigate the Turán exponent for $1$-subdivisions of graphs that are neither bipartite nor complete. Specifically, we establish an upper bound on the Turán 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 $4\le s\le t$. 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.

19 pages, 1 figure