Proof-critical dependence graph
Subgraphs versus Orientations: Infinite families of equidistributions
Full statement-restricted recursive proof-dependence graph for arXiv:2605.16028v1. Theorem 3.1 and its main enumerative corollaries are proved locally and independently verified. The equivalence-class family in Theorem 4.5 is not correct under the literal Section 2 digraph definitions: an arc-set cocycle cannot contain a 0-way edge, while reciprocal arcs of one 2-way nonloop edge form a digraph 2-cycle. Replacing those notions by the standard underlying-edge-simple potential cocycle/cycle notions makes Definition 4.4 meaningful and verifies the intended theorem. The chosen proof chain for Fact 4.10 uses Backman's graph-specific, self-contained cut/cycle decomposition; the independently sufficient BBY and Ding routes were inspected but are excluded as alternative proofs rather than drawn as parallel mandatory parents. Flajolet–Sedgewick is the terminal book source for Corollary 3.7. Historical, terminology, comparison, and locally reproved citations are excluded. Example 3.4 additionally needs X and Y nonempty, and two informal specializations need G connected.
Graph scope3 nodes2 proof-critical linksChecked August 31, 2026
Oriented proof graph
Dependence map
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- Solid arrow: headline proof lineage
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Evidence ledger
Proof-critical links
Partial Graph Orientations and the Tutte Polynomial→Subgraphs versus Orientations: Infinite families of equidistributionsHeadline lineageSelf-contained endpoint
Backman supplies a graph-specific, internally proved route. Apply its symmetric-difference decomposition to an orientation changed by reversing and then : because the cycles or cuts have no common arc, every opposite-arc overlap is reversed twice and cancels, while the edges in are reversed once. Backman's cut proof uses a Laplacian induction and its cycle proof is the dual argument.
Citation location: Focal PDF p. 11, citation [3]; Backman Theorems 3.2 and 3.4, PDF pp. 6–7.- Fact 4.10A conformal {0,±1}-difference of two directed cycles or cocycles decomposes into edge-disjoint directed cycles or cocycles.
Analytic Combinatorics→Subgraphs versus Orientations: Infinite families of equidistributionsHeadline lineageSource-chain endpoint
The labelled sequence and labelled set constructions justify the reciprocal and exponential generating-function transformations in the focal proof.
Citation location: Focal proof of Corollary 3.7, PDF pp. 5–6, citation [9]; Analytic Combinatorics, Chapter II §2, Theorem II.1, printed pp. 102–104; graph-set exponential formula in Example II.15, printed p. 138.- Corollary 3.7Generating-function relation between strongly connected tournaments and strongly connected simple digraphs.