Tur\'an number of the odd-ballooning of complete bipartite graphs

TL;DR

This paper determines the exact Turán number for the odd-ballooning of complete bipartite graphs $K_{s,t}$ with certain conditions, extending previous results to a broader class of graphs.

Contribution

It provides the first exact Turán number calculation for the odd-ballooning of $K_{s,t}$ with odd cycles of length at least five, for a wide range of parameters.

Findings

Exact Turán number for odd-ballooning of $K_{s,t}$ with $2 \\leq s \\leq t$, excluding $(2,2)$ and $(2,3)$.

Results apply when each odd cycle has length at least five.

Extends previous work from cycles, paths, and specific trees to bipartite graphs.

Abstract

Given a graph $L$, the Tur\'an number $\textrm{ex}(n,L)$ is the maximum possible number of edges in an $n$-vertex $L$-free graph. The study of Tur\'an number of graphs is a central topic in extremal graph theory. Although the celebrated Erd\H{o}s-Stone-Simonovits theorem gives the asymptotic value of $\textrm{ex}(n,L)$ for nonbipartite $L$, it is challenging in general to determine the exact value of $\textrm{ex}(n,L)$ for $\chi(L) \geq 3$. The odd-ballooning of $H$ is a graph such that each edge of $H$ is replaced by an odd cycle and all new vertices of odd cycles are distinct. Here the length of odd cycles is not necessarily equal. The exact value of Tur\'an number of the odd-ballooning of $H$ is previously known for $H$ being a cycle, a path, a tree with assumptions, and $K_{2,3}$. In this paper, we manage to obtain the exact value of Tur\'an number of the odd-ballooning of $K_{s,t}$ with $2\leq s \leq t$, where $(s,t) \not \in \{(2,2),(2,3)\} $ and each odd cycle has length at least five.