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Positroids are 3-colorable

Lamar Chidiac, Winfried Hochstättler

Abstract

We show that every positroid of rank $r \geq 3$ has a good coline. Using the definition of the chromatic number of oriented matroid introduced by J.\ Nešetřil, R.\ Nickel, and W.~Hochstättler, this shows that every orientation of a positroid is 3-colorable.

Positroids are 3-colorable

Abstract

We show that every positroid of rank has a good coline. Using the definition of the chromatic number of oriented matroid introduced by J.\ Nešetřil, R.\ Nickel, and W.~Hochstättler, this shows that every orientation of a positroid is 3-colorable.

Paper Structure

This paper contains 6 sections, 8 theorems, 7 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Loops, coloops and parallel elements in a L-graphs
  4. Connectivity of positroids
  5. Main theorem and proof
  6. Conclusion

Key Result

Theorem 1.1

Simple positroids of rank at least $2$ are $3$-colorable.

Figures (8)

  • Figure 1: A L-diagram $D$
  • Figure 2: The corresponding L-graph obtained from $D$
  • Figure 3: The L-graph of a positroid in which the sources 1 and 4 are loops, and the sink 5 is a coloop.
  • Figure 4: The L-graph of a positroid in which {3,4} and {6,7} are two parallel pairs.
  • Figure 5: A L-diagram with an isolated block formed by one level: edges labelled by 4,5,6 and 7. The other isolated block consists of two levels: edges labelled by 1,2,3,8 and 9
  • ...and 3 more figures

Theorems & Definitions (25)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2: Gammoids
  • Definition 2.3: L-diagram
  • Definition 2.4: L-graph
  • Definition 2.5: Positroids
  • Remark 2.1: Rank of a subset of a positroid
  • Definition 2.6: Closure operator
  • Definition 2.7: Copoints and Colines
  • Proposition 4.1: Proposition 4.1.3 in oxleybook
  • ...and 15 more