SIAM Journal on Discrete Mathematics, Volume 34, Issue 2, Page 1290-1301, January 2020.
Answering a question of Häggkvist and Scott, Verstraëte proved that every sufficiently large graph with average degree at least $k^2+19k+10$ contains $k$ vertex-disjoint cycles of consecutive even lengths. He further conjectured that the same holds for every graph $G$ with average degree at least $k^2+3k+2$. In this paper we prove this conjecture for $k\geq 19$ when $G$ is sufficiently large. We also show that for any $\epsilon>0$ and large $k\geq k_\epsilon$, average degree at least $k^2+3k-2+\epsilon$ suffices, which is asymptotically tight for infinitely many graphs.

中文翻译：

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顶点不相交周期上的Verstraëte猜想

SIAM离散数学杂志，第34卷，第2期，第1290-1301页，2020年1月。Verstraëte
回答了Häggkvist和Scott的问题，证明每个平均度至少为$ k ^ 2 + 19k + 10 $的足够大的图都包含连续偶数长度的$ k $个顶点不相交的循环。他进一步推测，每张图$ G $的平均度至少为$ k ^ 2 + 3k + 2 $时，情况相同。在本文中，我们证明了当$ G $足够大时，对于$ k \ geq 19 $的猜想。我们还显示，对于任何$ε> 0 $和大$ k \ geq k_epsilon $，平均度至少为$ k ^ 2 + 3k-2 + \ epsilon $，对于无限多的图来说，其渐近收敛。