The Cost of Perfection for Matchings in Graphs

Perfect matchings and maximum weight matchings are two fundamental combinatorial structures. We consider the ratio between the maximum weight of a perfect matching and the maximum weight of a general matching. Motivated by the application in triangle meshes, where we seek to convert a triangulation into a quadrangulation by merging pairs of adjacent triangles, we focus on bridgeless cubic graphs. First, we characterize graphs that attain the extreme ratios. Second, we present a lower bound for all bridgeless cubic graphs. Finally, we present upper bounds for subclasses of bridgeless cubic graphs that are relevant to the application, such as planar, hamiltonian, and bipartite. Most of the bounds are tight, but for bipartite graphs an intriguing gap is left as open problem.