Matchings: Source, Goal and Faithful Companion

TL;DR

The paper revisits Tutte's landmark matching theorem, highlighting its primarily combinatorial nature despite algebraic tools like the Tutte matrix and Pfaffians. It then develops a unifying framework for degree-constrained subgraphs (H-factors) using the concepts of sponges and, more generally, jump systems, to capture a wide range of factor and parity constraints. Core contributions include a short combinatorial proof of Tutte's theorem, a parity- and bounds-based min–max theory for classical factors, and a polynomial-time approach to sponge deficiency via parity-factor problems. The jump-system extension generalizes these results beyond graphs, enabling efficient solutions for general deficiency problems, including path matchings and matroid parity, thereby linking classical matching theory with factor theory, bidirected graphs, and algebraic complexity. The work provides a cohesive, algorithmically tractable path from Tutte's original insights to broad generalizations and applications in combinatorial optimization.

Abstract

Matchings were among the earliest motivations for graph theory. They subsequently remained a central goal, inspiring the development of new 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 legendary milestone in this trajectory is W. T. Tutte, "The factorization of linear graphs," J. Lond. Math. Soc. (1), 22, no. 2, (1947), 107-111, which firmly embedded graph theory through matchings into the body of classical mathematics, in particular, linear algebra and polynomials. In this note we revisit this article presenting its original content, sketching some aspects of its impact until some recent progress, and trace one of its subsequent lines of development finally leading to a new contribution answering an open challenge and extending known results.

Figures (6)

• Figure 1: Extracts from Tutte_book, entitled"Graphs within Graphs"
• Figure 2: Alternating paths for the Combinatorial version (of Lemma \ref{['lem:Jacobi']}) of Tutte's proof
• Figure 3: $M_u$:red(thick); $M_v$:blue(thin); $uv$:green; left: before exchange; right: after augmenting
• Figure 4: Distance function violating the triangle inequality, the subpath property, etc
• Figure 5: left: path; middle: circuit; right: failure of transitivity but rescue by Lemma \ref{['lem:trans']}

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Lemma 1.1
• Lemma 1.2
• Lemma 1.3
• Lemma 1.4
• Lemma 1.5
• Theorem 1.3
• Theorem 2.1
• Lemma 2.1