Maximum Induced Matchings of Random Regular Graphs

An induced matching of a graph G = (V,E) is a matching ${\mathcal M}$ such that no two edges of ${\mathcal M}$ are joined by an edge of E/${\mathcal M}$ In general, the problem of finding a maximum induced matching of a graph is known to be NP-hard. In random d-regular graphs, the problem of finding a maximum induced matching has been studied for d ∈ {3, 4, ..., 10 }. This was due to Duckworth et al.(2002) where they gave the asymptotically almost sure lower bounds and upper bonds on the size of maximum induced matchings in such graphs. The asymptotically almost sure lower bounds were achieved by analysing a degree-greedy algorithm using the differential equation method, whilst the asymptotically almost sure upper bounds were obtained by a direct expectation argument. In this paper, using the small subgraph conditioning method, we will show the asymptotically almost sure existence of an induced matching of certain size in random d-regular graphs, for d ∈ {3,4, 5}. This result improves the known asymptotically almost sure lower bound obtained by Duckworth et al.(2002).