Matchings: Source, goal, and faithful companion

Abstract Matchings were among the earliest motivations for graph theory. They subsequently remained a central goal, inspiring the development of new mathematical tools that went well beyond problems directly concerning matchings. These tools proved widely applicable, accompanying the growth of graph theory over the past century. A milestone in this trajectory is The factorisation of linear graphs , which firmly embedded graph theory—through matchings—into the body of classical mathematics, in particular, linear algebra and polynomials. We revisit this article by Tutte, presenting its original content, sketching some aspects of its impact until some recent progress, and tracing one of its subsequent lines of development, finally leading to a new contribution that answers an open challenge and extends known results.