Cycles of given size in a dense graph

TL;DR

This paper extends a classical graph theory result by proving that dense graphs with sufficient average degree contain multiple large, disjoint cycles, with bounds that are proven to be optimal in certain cases.

Contribution

The authors generalize a known theorem to show the existence of multiple large, disjoint cycles in dense graphs with a sharp bound for specific cycle sizes.

Findings

Graphs with average degree ≥ (4/3)kr contain k disjoint cycles of size at least r

The bound is proven to be sharp when r=3

The result extends previous work by Corrádi and Hajnal

Abstract

We generalise a result of Corr\'{a}di and Hajnal and show that every graph with average degree at least $\tfrac{4}{3}kr$ contains $k$ vertex disjoint cycles, each of order at least $r$, as long as $k \geq 6$. This bound is sharp when $r=3$.