Petersen's theorem is a classic result in matching theory from 1891, stating that every 3-regular bridgeless graph has a perfect matching. Our work explores efficient algorithms for finding perfect matchings in such graphs. Previously, the only relevant matching algorithms were for general graphs, and the fastest algorithm ran in O(n^3/2) time for 3-regular graphs. We have developed an O(n log^4 n)-time algorithm for perfect matching in a 3-regular bridgeless graph. When the graph is also planar, we have as the main result of our paper an optimal O(n)-time algorithm. We present three applications of this result: terrain guarding, adaptive mesh refinement, and quadrangulation.

Let G be a graph and T a set of vertices. A T-path in G is a path that begins and ends in T, and none of its internal vertices are contained in T. We define a T-path covering to be a union of vertex-disjoint T-paths spanning all of T. Concentrating on graphs that are tough (the removal of any nonempty set X of vertices yields at most |X | components), we completely characterize the edges that are contained in some T-path covering. Our main tool is Mader’s S-paths theorem. A corollary of our result is that each edge of a k-regular k-edge-connected graph (k ≥ 2) is contained in a T-path covering. This is, in a sense, a best possible counterpart of the result of Plesník that every edge of a k-regular (k − 1)edge-connected graph of even order is contained in a 1-factor.

By Petersen’s theorem, a bridgeless cubic graph has a 2-factor. H. Fleischner extended this result to bridgeless graphs of minimum degree at least three by showing that every such graph has a spanning even subgraph. Our main result is that, under the stronger hypothesis of 3-edge-connectivity, we can find a spanning even subgraph in which every component has at least five vertices. We show that this is in some sense best possible by constructing an infinite family of 3-edge-connected graphs in which every spanning even subgraph has a 5-cycle as a component.

Lovász and Plummer conjectured in the 1970’s that cubic bridgeless graphs have exponentially many perfect matchings. This conjecture has been verified for bipartite graphs by Voorhoeve in 1979, and for planar graphs by Chudnovsky and Seymour in 2008, but in general only linear bounds are known. In this paper, we provide the first superlinear bound in the general case.