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The local proofs are reconstructed on the result pages. The following results retain the external status they have in the 1965 source. The original proofs of these separate inputs are not counted here.

Finite representatives and flow

Section 2, printed pp. 148–149, uses:

• Hall's theorem. A finite indexed family $(A_i)_{i\in I}$ of finite sets has distinct representatives if and only if $|\bigcup_{i\in J}A_i|\ge|J|$ for every $J\subseteq I$. The complete original proof is compiled as Hall (1935), Theorem 1. The separate König min–max input below remains external.
• König's theorem. In a finite bipartite graph, the maximum cardinality of a matching equals the minimum cardinality of a vertex cover. A complete proof as a specialization of odd-set matching duality is compiled in Edmonds (1965), Section 5.9. It remains an external input to the present source.

These are the inputs to the Hall/König comparison. König's theorem also gives the rank reformulation of the network maximum.

Section 3, printed pp. 149–150, cites L. R. Ford, Jr., and D. R. Fulkerson, Flows in Networks (Princeton University Press, 1962), reference [3], for:

Integral max-flow/min-cut. In a finite directed network with nonnegative integer capacities and designated source and sink, the maximum feasible flow value equals the minimum capacity of a source–sink cut, and an integral maximum flow exists. An integral flow is a sum of unit source–sink paths and cycles; cycles can be discarded. The network used here is acyclic.

For the layered network used here, the flow equality and integral-optimum assertions are proved in Ford–Fulkerson (1957). Its terminal-direction assumptions hold because the source has only outgoing arcs and the sink has only incoming arcs. This separate original proof does not reproduce the 1962 book or its general decomposition theorem.

The source assigns infinite capacities to membership arcs. Our proof replaces them by $|E|+1$ and proves that a minimum cut never uses one, so the finite-capacity theorem suffices. Zero capacities are permitted, or equivalently their arcs may be deleted. No numerical flow computation or downloaded code is needed for the proof.

Edmonds's matching structure

Section 6, printed p. 153, invokes J. Edmonds, Paths, Trees and Flowers, Canadian Journal of Mathematics 17 (1965), 449–467, Section 6, reference [7]. The exact existence interface imported by the target paper is:

Let $G$ be a finite loopless graph and let $J$ be the set of vertices covered by every maximum-cardinality matching. The connected components $O_i$ of $G-J$ have odd orders $2r_i+1$. Let $Q$ be exactly the vertices of $J$ adjacent to $G-J$. There exists a maximum matching with $r_i$ edges internal to each $O_i$ and one edge from every $q\in Q$ to $G-J$. In the bipartite case every $O_i$ is a singleton.

The source states the stronger conclusion for every maximum matching and explicitly distinguishes it from the existence version attributed to [7]. The target paper's local strengthening and all lifting arguments are written in the matching-to-transversal proof. The original [7] proof is compiled separately in Edmonds's Theorem 6.2 and the resulting matching-decomposition interface. Those pages prove the stronger all-maximum and arbitrary-exposure claims. They remain external dependencies of this source unit and are not counted as same-paper proofs here.

Background and related source interfaces

References [1] and [2] are Edmonds's Minimum Partition of a Matroid into Independent Subsets and On Lehman's Switching Game and a Theorem of Tutte and Nash-Williams, both in the earlier 1965 issue of this journal. They provide historical context, but the present paper's different-matroid augmentation argument is fully reconstructed in Theorem 1c, with its local circuit and span lemmas. It is not replaced by a citation to either earlier paper.

The first-page acknowledgment reports that C. St. J. A. Nash-Williams obtained Theorems 1c, 2c, 1d and 2d by another method. That correspondence and its different proof are not printed here and have not been reconstructed in this unit.

Higgins's disjoint-transversal theorem and Tutte's graph decomposition and matroid papers are comparison references. Their original proofs and alternative theorem formulations are not claimed complete here. The elementary forest rank calculation needed to state the graphic specialization is supplied locally.

The note added in proof, printed p. 153, credits the earlier vector-space partition cases to Alfred Horn (1955) and R. Rado (1962). It also names the 1949 abstract-independence framework. The canonical Rado finite-rank page is a related interface, not a required external proof for our deductions from the 1965 independent-set axioms. The arbitrary-index Rado representative theorem uses its separate finite 1942 input. This unit does not silently replace that theorem or close its acquisition gap.

All sets in the 1965 proof are finite. No compactness, transfinite recursion, infinite-cardinality comparison, or infinite matroid extension is used. No source-specific formal proof or local formal build was checked for this unit.