Structure of the largest subgraphs of [formula omitted] with a given matching number

Abstract

This paper examines the structure of the largest subgraphs of the Erdős–Rényi random graph, Gn,p, with a given matching number. This extends a result of Erdős and Gallai who, in 1959, gave a classification of the structures of the largest subgraphs of Kn with a given matching number. We show that their result extends to Gn,p with high probability when p≥8lnnn or p≪1n, but that it does not extend (again with high probability) when 4ln(2e)n<p<lnn3n.