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A note on embracing exchange sequences in oriented matroids

Kristóf Bérczi, Benedek Nádor

TL;DR

This work generalizes a convex-geometry problem about transforming 0-embracing simplices into oriented matroid theory by introducing $e$-embracing bases and the $e$-embracing exchange distance. It conjectures a universal upper bound equal to the rank $r$ (recovering the geometric $d+1$ bound in the original setting) and proves this bound for oriented graphic matroids arising from directed graphs, with a monotone exchange sequence of length at most $r$. The paper situates the conjecture within classical symmetric-exchange problems (Gabow, White, Hamidoune) and demonstrates that certain strengthening to symmetric exchanges is impossible in general. By connecting convex geometry, matroid theory, and graph orientations, the results provide both constructive exchange procedures and inherent limitations for oriented matroids.

Abstract

An open problem in convex geometry asks whether two simplices $A,B\subseteq\mathbb{R}^d$, both containing the origin in their convex hulls, admit a polynomial-length sequence of vertex exchanges transforming $A$ into $B$ while maintaining the origin in the convex hull throughout. We propose a matroidal generalization of the problem to oriented matroids, concerning exchange sequences between bases under sign constraints on elements appearing in certain fundamental circuits. We formulate a conjecture on the minimum length of such a sequence, and prove it for oriented graphic matroids of directed graphs. We also study connections between our conjecture and several long-standing open problems on exchange sequences between pairs of bases in unoriented matroids.

A note on embracing exchange sequences in oriented matroids

TL;DR

This work generalizes a convex-geometry problem about transforming 0-embracing simplices into oriented matroid theory by introducing -embracing bases and the -embracing exchange distance. It conjectures a universal upper bound equal to the rank (recovering the geometric bound in the original setting) and proves this bound for oriented graphic matroids arising from directed graphs, with a monotone exchange sequence of length at most . The paper situates the conjecture within classical symmetric-exchange problems (Gabow, White, Hamidoune) and demonstrates that certain strengthening to symmetric exchanges is impossible in general. By connecting convex geometry, matroid theory, and graph orientations, the results provide both constructive exchange procedures and inherent limitations for oriented matroids.

Abstract

An open problem in convex geometry asks whether two simplices , both containing the origin in their convex hulls, admit a polynomial-length sequence of vertex exchanges transforming into while maintaining the origin in the convex hull throughout. We propose a matroidal generalization of the problem to oriented matroids, concerning exchange sequences between bases under sign constraints on elements appearing in certain fundamental circuits. We formulate a conjecture on the minimum length of such a sequence, and prove it for oriented graphic matroids of directed graphs. We also study connections between our conjecture and several long-standing open problems on exchange sequences between pairs of bases in unoriented matroids.

Paper Structure

This paper contains 9 sections, 1 theorem, 2 figures.

Table of Contents

  1. Introduction
  2. Basic notation
  3. General notation.
  4. Oriented matroids.
  5. Directed graphs.
  6. Matroidal extension
  7. Directed graphs
  8. Matroid exchange analogues
  9. Acknowledgement.

Key Result

Theorem 2.2

Let $D=(V,E)$ be a directed graph and $s,t\in V$. Suppose $A$ and $B$ are $st$-embracing spanning trees of $D$. Then there exist $st$-embracing spanning trees $A=T_0,T_1,\dots,T_{q-1},T_q=B$ with $q\le n-1$ such that $|T_{i-1}\triangle T_i|=2$ and $|T_i\cap B|\ge |T_{i-1}\cap B|$ for all $i\in[q]$.

Figures (2)

  • Figure 1: Illustration of Example \ref{['ex:graphic']}. Red and blue arcs correspond to $A$ and $B$, respectively. Note that $uw$ and $vw$ are contained in both spanning trees.
  • Figure 2: Illustration of Example \ref{['ex:symmetric']}. Red and blue triangles correspond to simplices $A$ and $B$, respectively. The only possible symmetric exchange uses $v$ and $y$.

Theorems & Definitions (9)

  • Conjecture 1.1: Caoduro, Khodamoradi, Paat, Shepherd
  • Conjecture 1.2
  • Example 2.1
  • Theorem 2.2
  • proof
  • Claim 2.3
  • proof
  • Conjecture 3.1: Hamidoune
  • Example 3.2