On the Tur\'{a}n number of $K_m \vee C_{2k-1}$

TL;DR

This paper determines the Turán number for the graph formed by the join of a complete graph and an odd cycle, providing an exact formula for sufficiently large graphs.

Contribution

It proves the Turán number for the join of $K_m$ and $C_{2k-1}$ for large enough graph sizes, extending known results in extremal graph theory.

Findings

Exact Turán number formula for $K_m \/ C_{2k-1}$

Identification of the threshold size for the formula's validity

Extension of classical extremal graph results

Abstract

Given a graph $H$ and a positive integer $n$, the Tur\'{a}n number of $H$ for the order $n$, denoted $ex(n,H)$, is the maximum size of a simple graph of order $n$ not containing $H$ as a subgraph. Given graphs $G$ and $H$, the notation $G \vee H$ means the joint of $G$ and $H$. $\chi(G)$ denotes the chromatic number of a graph $G$. Since $\chi(K_m \vee C_{2k-1})=m+3$ and there is an edge $e\in E(K_m \vee C_{2k-1})$ such that $\chi(K_m \vee C_{2k-1}-e)= m+2$, by the Simonovits theorem, $ex(n, K_m \vee C_{2k-1}) = \lfloor \frac{(m+1)n^2}{2(m+2)}\rfloor$ for sufficiently large $n$. In this paper, we prove that $2(m+2)k-3(m+2)-1$ is large enough for $n$.