Extremal graphs for odd-ballooning of bipartite graphs

TL;DR

This paper investigates the extremal number of edges in large graphs that avoid containing odd-ballooned bipartite graphs, establishing precise bounds for these Turán numbers when the cycles are sufficiently long.

Contribution

It determines the Turán numbers for odd-ballooning of bipartite graphs with cycle length at least five, extending extremal graph theory to new classes of graphs.

Findings

Established the range of Turán numbers for odd-ballooning of bipartite graphs for t ≥ 5.

Derived Turán numbers for odd-ballooning of stars, paths, and even cycles.

Provided new extremal bounds for graphs avoiding odd-cycle expansions of bipartite graphs.

Abstract

Given a graph $H$ and an odd integer $t$ ($t\geq 3$), the odd-ballooning of $H$, denoted by $H(t)$, is the graph obtained from replacing each edge of $H$ by an odd cycle of length at least $t$ where the new vertices of the cycles are all distinct. In this paper, we determine the range of Tur\'{a}n numbers for odd-ballooning of bipartite graphs when $t\geq 5$. As applications, we may deduce the Tur\'{a}n numbers for odd-ballooning of stars, paths and even cycles.