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：我们研究在图中寻找最大规模无三角形$2$-匹配的问题。$2$-匹配是边的任意子集，使得每个顶点至多与该子集中的两条边相关联。我们提出了一种针对该问题的快速组合算法。相比于Hartvigsen于1984年提出的极为复杂的结果，我们的算法及其分析显著简捷。在算法设计中，我们引入了若干新概念。已有研究证明，对于任何非最大的无三角形$2$-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，如何高效寻找此类路径此前未知。一项新发现是：增广路径$P$的搜索可限定于所谓的“顺应”路径，这类路径对$P \cup M$中包含的任意三角形$t$的访问次数有限。为了找到顺应路径（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。此操作可视为使用“小工具”，其中某些边对被断开连接。为能断开两条边，我们引入“半边”概念。非正式地，边$e$的“半边”是$e$恰好包含其一个端点的半段。这是半边的又一新颖应用——此前半边被用于旅行商问题及其他匹配问题。此外，小工具在任意增广阶段并非固定不变，而是根据顺应路径当前发现的可达状态动态调整。