bibkey: schmerltrotter1993critical authors: James H. Schmerl and William T. Trotter year: 1993 title: Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures doi: 10.1016/0012-365X(93)90516-V url: https://trotter.math.gatech.edu/papers/83.pdf claim: The classical odd-order critical tournament family T_r^(3) is a transitive chain together with one pivot oriented toward alternating chain vertices and away from the others. strata_touched:

• D5/S3/ConceptDynamics/GraphIrregularity/LinkIrregularTournamentExistence license: citation-only triage: anchor

Critical binary relational structures

On page 198, Schmerl and Trotter define the tournament family denoted T_r^(3). Its vertices consist of a linearly ordered chain and one additional vertex whose incident arcs alternate along that chain. After changing from one-based to zero-based chain labels, the additional vertex points to the even labels and receives arcs from the odd labels. The repository uses this classical W family at odd orders at least seven and adds one sink to the preceding odd-order W at even orders at least eight; both constructions have the displayed alternating-pivot relation. The even-order extension is not identified here as a classical critically indecomposable W tournament.

The paper studies critical indecomposability. No exact all-order link-irregular tournament existence theorem was located in its inspected statements; text extraction elsewhere in the PDF had imperfect OCR. In the repository proof, after deleting a chain vertex, the surviving pivot is the unique vertex whose further deletion leaves a transitive tournament: three disjoint adjacent chain pairs ensure that deleting any other vertex leaves an intact triangle. A chain-deleted-card isomorphism therefore fixes the pivot and then the chain ranks, exposing opposite pivot-edge parity at the smaller deleted rank. The pivot-deleted card itself is transitive, whereas each chain-deleted card is not.

Verified locator

• DOI: https://doi.org/10.1016/0012-365X(93)90516-V
• Author-hosted PDF: https://trotter.math.gatech.edu/papers/83.pdf