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.

We study extremal and structural problems in regular graphs involving various parameters. In Chapter 2, we obtain the best lower bound for the matching number over n-vertex connected regular graphs in terms of edge-connectedness and determine when the matching number is minimized. We also establish the best upper bound for the number of cut-edges over n-vertex connected odd regular graphs and determine when the number of cut-edges is maximized. In addition, there is a relationship between the matching number and the total domination number in regular graphs. In Chapter 3, we explore the relationship between eigenvalue and matching number in regular graphs. We give a condition on an appropriate eigenvalue that guarantees a lower bound for the matching number of a l-edge-connected d-regular graph, when l ≤ d − 2. We also study what is the weakest hypothesis on the second largest eigenvalue λ2 for a d-regular graph G to guarantee that G is l-edge-connected. In Chapter 4, we study several extremal problems for regular graphs, including the Chinese postman problem, the path cover number, the average edge-connectivity, and the number of perfect matchings. In Chapter 5, we study an r-dynamic coloring problem and give the relationship between the r-dynamic chromatic number and the chromatic number in regular graphs. We also study r-dynichromatic number of the cartesian product of paths and cycles.