A Density Version of the Corradi-Hajnal Theorem

TL;DR

This paper establishes a density condition on graphs that guarantees the existence of k disjoint cycles, extending the Corradi-Hajnal theorem with optimal edge count bounds.

Contribution

It introduces a density version of the Corradi-Hajnal theorem, providing the best possible edge threshold for k disjoint cycles in large graphs.

Findings

Graphs with sufficiently many edges contain k disjoint cycles.

The edge bounds are proven to be optimal.

The result generalizes and strengthens previous theorems.

Abstract

For every positive integer $k$, we show that every graph of order $n$ at least $3k$ with more than $$\max\{{2k-1\choose 2}+(2k-1)(n-(2k-1)),{3k-1\choose 2}+(n-(3k-1))\}$$ edges has $k$ vertex disjoint cycles, which is a best possible density version of a theorem of Corr\'{a}di and Hajnal.