Generalized Turan number for the edge blow-up graph

TL;DR

This paper establishes precise extremal bounds for specific edge blow-up graphs, advancing understanding of Turan numbers for complex graph structures.

Contribution

It provides sharp upper bounds for $ex(n,K_3,C_3^3)$ and exact values for $ex(n,K_3,P_3^3)$, identifying extremal graphs.

Findings

Sharp upper bounds for $ex(n,K_3,C_3^3)$

Exact value for $ex(n,K_3,P_3^3)$

Characterization of extremal graphs

Abstract

Let $H$ be a graph and $p$ be an integer. The edge blow-up $H^p$ of $H$ is the graph obtained from replacing each edge in $H$ by a copy of $K_p$ where the new vertices of the cliques are all distinct. Let $C_k$ and $P_k$ denote the cycle and path of length $k$, respectively. In this paper, we find sharp upper bounds for $ex(n,K_3,C_3^3)$ and the exact value for $ ex(n,K_3,P_3^3)$ and determine the graphs attaining these bounds.