We consider the problem of finding a maximum size triangle-free $2$-matching in a graph $G$. A $2$-matching is any subset of the edges such that each vertex is incident to at most two edges from the subset. We present a fast combinatorial algorithm for the problem. Our algorithm and its analysis are dramatically simpler than the very complicated result by Hartvigsen from 1984. In the design of this algorithm we use several new concepts. It has been proven before that for any triangle-free $2$-matching $M$ which is not maximum the graph contains an $M$-augmenting path, whose application to $M$ results in a bigger triangle-free $2$-matching. It was not known how to efficiently find such a path. A new observation is that the search for an augmenting path $P$ can be restricted to so-called {\em amenable} paths that go through any triangle $t$ contained in $P \cup M$ a limited number of times. To find an augmenting path that is amenable and hence whose application does not create any triangle we forbid some edges to be followed by certain others. This operation can be thought of as using gadgets, in which some pairs of edges get disconnected. To be able to disconnect two edges we employ {\em half-edges}. A {\em half-edge} of edge $e$ is, informally speaking, a half of $e$ containing exactly one of its endpoints. This is another novel application of half-edges which were previously used for TSP and other matching problems. Additionally, gadgets are not fixed during any augmentation phase, but are dynamically changing according to the currently discovered state of reachability by amenable paths.

翻译：我们考虑在图中寻找最大无三角形$2$-匹配的问题。$2$-匹配是指边集的任意子集，其中每个顶点至多与该子集中的两条边相关联。我们提出了一种快速组合算法来解决该问题。该算法及其分析比Hartvigsen在1984年提出的极为复杂的结果要简单得多。在算法设计中，我们使用了若干新概念。已有证明表明，对于任意非最大的无三角形$2$-匹配$M$，图中存在一条$M$-增广路径，将其应用于$M$可得到更大的无三角形$2$-匹配。然而，如何高效地找到这样的路径此前并不清楚。一个新发现是，增广路径$P$的搜索可以限制在所谓的“可接纳”路径中，这些路径对$P \cup M$中包含的任何三角形$t$的遍历次数有限。为找到可接纳路径（从而确保其应用不会产生任何三角形），我们禁止某些边被特定边跟随。这一操作可视为使用“小装置”，其中某些边对之间被断开连接。为了实现边的断开，我们采用了“半边”概念。非正式地说，边$e$的“半边”是包含其一个端点的半条边。这是半边的又一新颖应用，此前半边已被用于旅行商问题及其他匹配问题。此外，在任意增广阶段，小装置并非固定不变，而是根据当前通过可接纳路径发现的可达状态动态变化。