Indicated coloring of matroids

TL;DR

This paper proves that the indicated chromatic number of a loopless matroid, defined via a game where Alice indicates elements and Bob colors them properly, equals the matroid's chromatic number, establishing a key equivalence.

Contribution

It introduces and proves the equality of the indicated chromatic number and the traditional chromatic number for matroids, connecting game-theoretic and classical coloring concepts.

Findings

The indicated chromatic number equals the chromatic number for matroids.

The game-theoretic variant aligns with the classical coloring parameter.

The result holds for all loopless matroids.

Abstract

A coloring of a matroid is proper if elements of the same color form an independent set. For a loopless matroid M, its chromatic number \chi(M) is the minimum number of colors that suffices to color properly the ground set E of M. In this note we study a game-theoretic variant of this parameter proposed by Grytczuk. Suppose that in each round of the game Alice indicates an uncolored yet element e of E, then Bob colors it using a color from a fixed set of colors C. The rule Bob has to obey is that it is a proper coloring. The game ends if the whole matroid has been colored or if Bob can not color e using any color of C. Alice wins in the first case, while Bob in the second. The minimum size of the set of colors C for which Alice has a winning strategy is called the indicated chromatic number of M, denoted by \chi_i(M). We prove that \chi_i(M)=\chi(M).