Small Maximal Matchings of Random Cubic Graphs

Abstract

We consider the expected size of a smallest maximalmatching of cubic graphs. Firstly, we present a randomized greedy algorithm for finding a small maximal matching of cubic graphs.We analyze the averagecase performance of this heuristic on random n-vertex cubic graphs using differential equations. In this way, we prove that the expected size of the maximalmatching returned by the algorithm is asymptotically almost surely (a.a.s.) less than 0.34623n. We also give an existence proof which shows that the size of a smallest maximal matching of a random n-vertex cubic graph is a.a.s. less than 0.3214n. It is known that the size of a smallest maximal matching of a random n-vertex cubic graph is a.a.s. larger than 0.3158n. Show more