A Note on the Graphs with Given Small Matching Number

Abstract

In this note, we characterize graphs with a given small matching number. Specifically, we characterize graphs with minimum degree at least \(2\) and matching number at most \(3\). The characterization when the matching number is at most \(2\) strengthens the result of Lai and Yan’s that characterized the non-supereulerian \(2\)-edge connected graphs with matching at most \(2\). Furthermore, the characterization of graphs with matching number at most \(3\) addresses a conjecture of Lai and Yan in [SuperEulerian graphs and matchings, Applied Mathematics Letters 24 (2011) 1867-1869].