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On $k$-neighborly reorientations of oriented matroids

Rangel Hernández-Ortiz, Kolja Knauer, Luis Pedro Montejano

Abstract

We study the existence and the number of $k$-neighborly reorientations of an oriented matroid. This leads to $k$-variants of McMullen's problem and Roudneff's conjecture, the case $k=1$ being the original statements on complete cells in arrangements. Adding to results of Larman and García-Colín, we provide new bounds on the $k$-McMullen's problem and prove the conjecture for several ranks and $k$ by computer. Further, we show that $k$-Roudneff's conjecture for fixed rank and $k$ reduces to a finite case analyse. As a consequence we prove the conjecture for odd rank $r$ and $k=\frac{r-1}{2}$ as well as for rank $6$ and $k=2$ with the aid of the computer.

On $k$-neighborly reorientations of oriented matroids

Abstract

We study the existence and the number of -neighborly reorientations of an oriented matroid. This leads to -variants of McMullen's problem and Roudneff's conjecture, the case being the original statements on complete cells in arrangements. Adding to results of Larman and García-Colín, we provide new bounds on the -McMullen's problem and prove the conjecture for several ranks and by computer. Further, we show that -Roudneff's conjecture for fixed rank and reduces to a finite case analyse. As a consequence we prove the conjecture for odd rank and as well as for rank and with the aid of the computer.
Paper Structure (13 sections, 21 theorems, 29 equations, 7 figures)

This paper contains 13 sections, 21 theorems, 29 equations, 7 figures.

Table of Contents

  1. Introduction
  2. $k$-McMullen's problem
  3. $k$-Roudneff's conjecture
  4. Organization of the paper
  5. Results on the $k$-McMullen problem
  6. Results on the $k$-Roudneff conjecture
  7. Orthogonality and neighborliness
  8. The $o$-vector of $\mathcal{C}_r(n)$ and its tope graph
  9. Reducing $k$-Roudneff's conjecture to a finite case analysis and proving the case $k=\frac{r-1}{2}$
  10. $k$-McMullen's problem and $k$-Roudneff's conjecture for low ranks
  11. A computer program that obtains $o(\mathcal{M},k)$
  12. Results on $k$-McMullen's problem
  13. Results on $k$-Roudneff's conjecture

Key Result

Lemma 2.1

An oriented matroid $\mathcal{M}=(E,\mathcal{C})$ is $k$-neighborly if and only if every subset $F\subseteq E$ of size at most $k$ is a face.

Figures (7)

  • Figure 1: The graphs $B(3,5)$ and $B(3,4)$ corresponding to the tope graphs of $\mathcal{C}_3(5)$ and $\mathcal{C}_3(4)$, respectively. Red vertices correspond to $2$-orthogonal topes (i.e., $1$-neighborly topes) and adjacent topes between a blue edge, differ exactly in the fifth entry.
  • Figure 2: In the first example $j\in S^X\setminus H^X$, $j+1\in S^X\cap H^X$ and $j+2\in H^X\setminus S^X$. In the second example, $j,j+1,j+2\in S^{X}\setminus H^X$.
  • Figure 3: An example of the case $r+1-m$ odd in \ref{['lemmageneral:blocks-ortogonality']}.
  • Figure 4: An example of the case $|H^X_{<j}|\le \lfloor \frac{\mathbf{B}_o}{2}\rfloor$ of \ref{['lemmageneral:blocks-ortogonality2']}. Notice that $2=|H^X_{<j}|=|H^X_{<3}|=\lfloor\frac{\mathbf{B}_o}{2}\rfloor$, $H^X_{<j}=\{1,2\}$, $S^X=\{3,4,5,6,7,8,9\}$ and $H^X=\{1,2,3\}$.
  • Figure 5: An example of the case $|H^X_{<j}|> \lfloor \frac{\mathbf{B}_o}{2}\rfloor$ of \ref{['lemmageneral:blocks-ortogonality2']}. Notice that $|H^X_{<j}|=|H^X_{<6}|=|\{3,4,5\}|=3$, $\lfloor \frac{\mathbf{B}_o}{2}\rfloor=2$, $S^X=\{1,2,6,7,8,9\}$, $H^X=\{2,3,4,5,6\}$, $S^{X'}=\{1,2,9\}$, $H^{X'}=\{2,3,4,5,6,7,8,9\}$ and $S^X_{<j}=S^{X'}_{<j}=\{1\}$.
  • ...and 2 more figures

Theorems & Definitions (52)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Conjecture 1.4: McMullen 1972
  • Remark 1.6
  • proof
  • Conjecture 1.7: Roudneff 1991
  • Lemma 2.1
  • proof
  • Remark 2.2
  • ...and 42 more