New results on $k$-Roudneff's conjecture

TL;DR

The paper advances k-Roudneff's conjecture by establishing its validity for Lawrence oriented matroids at even rank r=2k+2 with large n, providing a general upper bound for k-neighborly reorientations in LO Ms, and proving asymptotically (as n→∞) that the conjecture holds for all oriented matroids. It introduces travel-based characterizations (plain travels, top/bottom travels) and the chessboard framework to count reorientation classes, and it employs computer-assisted proofs to verify several low-rank cases. The combination of structural results, upper bounds, and asymptotic analysis significantly strengthens confidence in the conjecture across broader classes of oriented matroids. Overall, the work blends combinatorial, computational, and asymptotic techniques to deepen understanding of k-neighborliness in matroid theory and its extremal reorientation behavior.

Abstract

In this paper we study the number of $k$-neighborly reorientations of an oriented matroid, leading to study $k$-Roudneff's conjecture, the case $k=1$ being the original statement conjectured in 1991. We first prove the conjecture for the family of Lawrence oriented matroids (LOMs) with even rank $r=2k+2$ and also for low ranks by computer. Next, we provide a general upper bound for the number of $k$-neighborly reorientations of any LOM. Finally, we prove that for any $k\ge 1$ and any oriented matroid on $n$ elements, $k$-Roudneff's conjecture holds asymptotically as $n\rightarrow \infty$ and thus giving more credit to the conjecture.

Figures (8)

• Figure 1: In black, the top and bottom travels of the matrix $A$ and in blue, a plain travel $P$ of $A$. The matrix $\mathcal{A}$ is obtaining by reorient column $2$ of $A$. Notice that $P$ is the top travel of $\mathcal{A}$.
• Figure 2: In black, the top and bottom travels of the matrices $A$ and $A_S$ and in blue, a plain travel $P$ of $A$. The matrix $A_S$ is obtaining by reorient columns $3$ and $5$ of $A$. The $TT$ and the $BT$ of $A_S$ are positive.
• Figure 3: If $a_{1,4}\in TT$, then depending on the signs of $a_{2,2}$, $a_{2,3}$ and $a_{2,4}$, we must reorient column $2$ or $3$. In the first example (matrix $A$), we must reorient column $2$ in order to make its top travel positive. If $a_{1,4}\notin TT$ (see matrix B), then we must reorient at most one column, for example column 4 (if necessary), in order to make the $TT$ positive.
• Figure 4: The sets $V_j$ and $H_j$.
• Figure 5: The situation of Lemma \ref{['Extesionk']}

Theorems & Definitions (60)

• Conjecture 1.1: Roudneff 1991
• Conjecture 1.2: $k$-Roudneff's conjecture
• Remark 1.3
• Remark 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• Lemma 2.5
• proof