Extremal graphs for edge blow-up of lollipops

TL;DR

This paper investigates the extremal graphs for the edge blow-up of lollipop graphs, focusing on cases where the parameter p equals 2 and 3, extending previous work on Turán numbers for such graph transformations.

Contribution

It provides new results on extremal graphs for the edge blow-up of lollipop graphs specifically for p=2 and p=3, filling gaps in the existing literature.

Findings

Characterization of extremal graphs for C_{k, }^(p+1) when p=2 and p=3.

Extension of Turán number results to new classes of graphs.

Identification of extremal configurations for specific edge blow-ups.

Abstract

Given a graph $H$ and an integer $p$ ($p\geq 2$), the edge blow-up $H^{p+1}$ of $H$ is the graph obtained from replacing each edge in $H$ by a clique of order $(p+1)$, where the new vertices of the cliques are all distinct. The Tur\'{a}n numbers for edge blow-up of matchings were first studied by Erd\H{o}s and Moon. Very recently some substantial progress of the extremal graphs for $H^{p+1}$ of larger $p$ has been made by Yuan. The range of Tur\'{a}n numbers for edge blow-up of all bipartite graphs when $p\geq 3$ and the exact Tur\'{a}n numbers for edge blow-up of all non-bipartite graphs when $p\geq \chi(H) +1$ has been determined by Yuan (2022), where $\chi(H)$ is the chromatic number of $H$. A lollipop $C_{k,\;\ell}$ is the graph obtained from a cycle $C_k$ by appending a path $P_{\ell+1}$ to one of its vertices. In this paper, we consider the extremal graphs for $C_{k,\;\ell}^{p+1}$ of the rest cases $p=2$ and $p=3$.