Abstract An even cycle decomposition of a graph is a partition of its edges into cycles of even length. In 2012, Markstrom conjectured that the line graph of every 2-connected cubic graph has an even cycle decomposition and proved this conjecture for cubic graphs with oddness at most 2. However, for 2-connected cubic graphs with oddness 2, Markstrom only considered these graphs with a chordless 2-factor. (A chordless 2-factor of a graph is a 2-factor consisting of only induced cycles.) In this paper, we first construct an infinite family of 2-connected cubic graphs with oddness 2 and without chordless 2-factors. We then give a complete proof of Markstrom’s result and further prove this conjecture for cubic graphs with oddness 4.

中文翻译：

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关于三次图线图的偶数循环分解

摘要 图的偶循环分解是将其边划分为偶数长度的循环。2012 年，Markstrom 猜想每一个 2-连通三次图的线图都有一个偶循环分解，并证明了这个猜想对于奇数最多为 2 的三次图。 然而，对于奇数为 2 的 2-连通三次图，Markstrom 只考虑了这些具有无弦 2 因子的图。（图的无弦二因数是仅由诱导圈组成的二因数。）在本文中，我们首先构造奇数为 2 且无无弦二因数的 2-连通三次图的无限族。然后我们给出了 Markstrom 结果的完整证明，并进一步证明了奇数为 4 的三次图的这个猜想。