Even cycles in dense graphs

TL;DR

This paper proves that dense graphs with certain minimum degree conditions contain disjoint unions of even cycles, unless they have a very particular structure, extending understanding of cycle embeddings in dense graphs.

Contribution

It establishes conditions under which dense graphs contain specified even cycles, identifying exceptions with a precise structural characterization.

Findings

Dense graphs contain specified even cycles under given degree conditions.

Identifies a unique structural exception to cycle containment.

Provides a threshold for the minimum degree ensuring cycle unions.

Abstract

We will show that for $\alpha>0$ there is $n_0$ such that if $G$ is a graph on $n\geq n_0$ vertices such that $\alpha n< \delta(G)< (n-1)/2$, then for every $n_1+n_2+\cdots +n_l= \delta(G)$, $G$ contains a disjoint union of $C_{2n_1},C_{2n_2}, \dots, C_{2n_l}$ unless $G$ has a very specific structure.