Extremal graphs for odd-ballooning of paths and stars

TL;DR

This paper determines the extremal graphs for the odd-ballooning of paths and stars for all odd cycle sizes greater than or equal to 3, extending previous results using Simonovits' progressive induction method.

Contribution

It provides a unified proof for the extremal graphs of both odd-ballooning of paths and stars for q ≥ 3, generalizing earlier specific cases.

Findings

Identifies extremal graphs for odd-ballooning of paths for q ≥ 3.

Identifies extremal graphs for odd-ballooning of stars for q ≥ 3.

Extends previous results to a broader class of graphs.

Abstract

The odd-ballooning of a graph $G$, denoted by $G_q$, is the graph obtained from replacing each edge in $G$ by a odd cycle of the same size where the new vertices of the odd cycles are all different. In 2002, Erd\"os et al. determined the extremal graphs of $k$-fan. In 2016, Hou et al. determined extremal graphs of the odd-ballooning of stars for $q\geqslant 5$. In 2020, Zhu et al. determined extremal graphs of the odd-ballooning of paths for $q\geqslant 3$. In this article, we use progressive induction lemma of Simonovits to determine the extremal graphs of both odd-ballooning of stars and odd-ballooning of paths for $q\geqslant 3$.