On the exact decomposition threshold for even cycles

TL;DR

This paper establishes exact minimum degree conditions for large graphs to have decompositions into even cycles, improving upon previous asymptotic results and confirming their optimality.

Contribution

It provides the precise minimum degree thresholds for decomposing large graphs into even cycles, including specific bounds for 4-cycles and longer even cycles, confirming their optimality.

Findings

Minimum degree $rac{2|G|}{3}-1$ guarantees $C_4$-decomposition.

Minimum degree $rac{|G|}{2}$ guarantees $C_{2k}$-decomposition for $k extgreater 3$.

Results are tight and extend to bipartite graphs and those with expansion properties.

Abstract

A graph $G$ has a $C_k$-decomposition if its edge set can be partitioned into cycles of length $k$. We show that if $\delta(G)\geq 2|G|/3-1$, then $G$ has a $C_4$-decomposition, and if $\delta(G)\geq |G|/2$, then $G$ has a $C_{2k}$-decomposition, where $k\in \mathbb{N}$ and $k\geq 4$ (we assume $G$ is large and satisfies necessary divisibility conditions). These minimum degree bounds are best possible and provide exact versions of asymptotic results obtained by Barber, K\"uhn, Lo and Osthus. In the process, we obtain asymptotic versions of these results when $G$ is bipartite or satisfies certain expansion properties.