English Translation of "Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre" by Dénes Kőnig

TL;DR

This work unifies determinant theory, set theory, and topology through graph theory by showing that regular bipartite graphs admit decompositions into first-degree factors, enabling combinatorial proofs of determinant properties. It translates determinant problems into graph-theoretic ones, proving that equal row/column sums imply nonzero determinant terms and, for unit nonzeros, a guaranteed number of nonzero terms, with connections to edge-coloring and the 4-color conjecture. The paper also extends these ideas to infinite/countable graphs, linking graph decompositions to cardinality arguments in set theory via reversible (1,$\nu$)-relations, and discusses the limitations and potential for generalization, including the role of prime-power degrees. Overall, it provides a foundational combinatorial framework for understanding determinant behavior and cardinality through graph decompositions.

Abstract

The presented work focuses on problems from determinant theory, set theory and topology. The term graph is the binding element that connects these problems. Graphs are distinguished by their geometrical simplicity, which helps in showing the equivalence between various seemingly unrelated problems, besides providing solutions to several open questions discussed here.

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