Reproduction and Pedagogical Contributions
Maximum Matching in Bipartite Graphs and its Generalizations
DOI:
https://doi.org/10.20873/uft.2675-3588.2026.v7n2.p21-28Keywords:
Maximum Matching, Tutte's Theorem, Dilworth's Theorem, Kameda-Munro, Graph Theory EducationAbstract
This paper presents a didactic study on the Maximum Matching problem in graphs, with an emphasis on bipartite graph structures and their generalizations. The objective is to reproduce fundamental results that move beyond the classical approach of König and Hall. To this end, we explore Tutte's Theorem, which conditions perfect matching on the analysis of odd components, and Dilworth's Theorem, which establishes a duality with partially ordered sets (posets). The methodology employs the analysis of proofs, utilizing techniques of reduction and decomposition, accompanied by illustrative examples and strategic visualizations. As central results, we demonstrate that Tutte's condition is the universal structural obstacle to perfect matching, and that Dilworth's equivalence establishes a rigorous reduction between poset decomposition and bipartite matching, enabling efficient polynomial-time solutions as exemplified by the works of Kameda and Munro. In conclusion, this study fills conceptual gaps and offers a significant pedagogical contribution, making the rigor of Graph Theory more accessible to undergraduate students and promoting a unified view on the existence of matchings and chain covers.
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Copyright (c) 2026 Vitória Milhomem Soares, Matheus de Sousa Silva, Daniel Martins da Silva, Tanilson Dias dos Santos
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