# A generalization of Bondy's pancyclicity theorem

## TL;DR

This paper extends Bondy's classical pancyclicity theorem by showing that a graph with minimum degree at least its bipartite independence number is either pancyclic or a complete bipartite graph, generalizing previous Hamiltonicity results.

## Contribution

It generalizes Bondy's theorem by establishing pancyclicity under a broader minimum degree condition involving the bipartite independence number.

## Key findings

- Graphs with minimum degree ≥ bipartite independence number are pancyclic or complete bipartite.
- Extends McDiarmid and Yolov's Hamiltonian condition to pancyclicity.
- Provides a unified condition encompassing Dirac's and Bondy's theorems.

## Abstract

The bipartite independence number of a graph $G$, denoted as $\tilde\alpha(G)$, is the minimal number $k$ such that there exist positive integers $a$ and $b$ with $a+b=k+1$ with the property that for any two sets $A,B\subseteq V(G)$ with $|A|=a$ and $|B|=b$, there is an edge between $A$ and $B$. McDiarmid and Yolov showed that if $\delta(G)\geq\tilde \alpha(G)$ then $G$ is Hamiltonian, extending the famous theorem of Dirac which states that if $\delta(G)\geq |G|/2$ then $G$ is Hamiltonian. In 1973, Bondy showed that, unless $G$ is a complete bipartite graph, Dirac's Hamiltonicity condition also implies pancyclicity, i.e., existence of cycles of all the lengths from $3$ up to $n$. In this paper we show that $\delta(G)\geq\tilde \alpha(G)$ implies that $G$ is pancyclic or that $G=K_{\frac{n}{2},\frac{n}{2}}$, thus extending the result of McDiarmid and Yolov, and generalizing the classic theorem of Bondy.