Transitive and Gallai colorings

TL;DR

This paper introduces transitive colorings as an analogue to Gallai colorings for acyclic directed graphs, generalizes these notions to Coxeter systems and matroids, and explores their combinatorial properties and enumerations.

Contribution

It generalizes Gallai and transitive colorings to broader algebraic structures like Coxeter systems and matroids, and establishes new counting formulas and symmetry properties.

Findings

Maximal number of colors equals matroid rank.

Number of colorings with at most k colors is polynomial in k.

Number of transitive 2-colorings equals the number of chambers in a hyperplane arrangement.

Abstract

A Gallai coloring of the complete graph is an edge-coloring with no rainbow triangle. This concept first appeared in the study of comparability graphs and anti-Ramsey theory. We introduce a transitive analogue for acyclic directed graphs, and generalize both notions to Coxeter systems, matroids and commutative algebras. It is shown that for any finite matroid (or oriented matroid), the maximal number of colors is equal to the matroid rank. This generalizes a result of Erd\H{o}s-Simonovits-S\'os for complete graphs. The number of Gallai (or transitive) colorings of the matroid that use at most $k$ colors is a polynomial in $k$. Also, for any acyclic oriented matroid, represented over the real numbers, the number of transitive colorings using at most 2 colors is equal to the number of chambers in the dual hyperplane arrangement. We count Gallai and transitive colorings of the root system of type A using the maximal number of colors, and show that, when equipped with a natural descent set map, the resulting quasisymmetric function is symmetric and Schur-positive.