Skip to content

Sequential Matroid Has Canonical Left And Right Ends

Claim/Theorem

Let \(M\) be a sequential matroid of rank and corank at least 3, and assume \(M\) is neither a wheel nor a whirl.

Then there exist distinct subsets

\[ L(M),\ R(M)\subseteq E(M) \]

such that for every sequential ordering \(\Sigma\) of \(E(M)\),

\[ \{L(M),R(M)\}=\{L(\Sigma),R(\Sigma)\}, \]

where the canonical end seen in any given sequential ordering has one of the following refined endpoint types:

  1. triangle end,
  2. triad end,
  3. segment end,
  4. cosegment end,
  5. fan end.

Equivalently, every non-exceptional sequential matroid has two canonical fixed ends shared by all sequential orderings. At the coarser level emphasized in the paper's abstract and Theorem 1.2, these fixed ends are controlled by maximal segment, maximal cosegment, or maximal fan templates; the triangle and triad cases are the 3-element boundary cases used in the finer Theorem 1.3 / 1.4 terminology.

For Conjecture 3, this means the remaining sequential cut-rank-2 loophole is not arbitrary path-width-three behavior. It already comes with a pair of canonical end-templates that every sequential ordering must respect.

Dependencies

  • None.

Conflicts/Gaps

  • The theorem excludes wheels and whirls, and assumes rank and corank at least 3.
  • The paper uses two closely related layers of terminology: Theorem 1.2 speaks in terms of maximal segments, maximal cosegments, and maximal fans, while Theorems 1.3 and 1.4 refine the small 3-element endpoint cases into triangle and triad ends.
  • It controls only the end structure of sequential orderings, not the full middle of the ordering.
  • Therefore it sharply constrains the sequential branch but does not yet exclude sequential low-rank obstructions from the relevant Quantum Tanner parity-check matroids.

Sources

  • 10.1016/j.ejc.2005.10.005 (Theorem 1.2 on p. 965; Theorems 1.3 and 1.4 on pp. 966-967)