The connection between the chromatic function and the Redei-Berge function

TL;DR

This paper explores the deep connection between the chromatic function of graphs and the Redei-Berge function of digraphs through their poset representations, revealing a reflection of their noncommutative generalizations.

Contribution

It demonstrates that the relationship between these functions is a reflection of their noncommutative counterparts, enabling property transfer and new results.

Findings

Derived the converse of Redei's theorem

Generalized the triple deletion property

Provided expressions for functions in special cases

Abstract

There is a natural way to assign both graph and digraph to every poset. Furthermore, any graph has its chromatic function, while any digraph has its Redei-Berge function. On the level of posets, these two functions are almost identical. Here, we prove that this connection is actually a reflection of the connection between the noncommutative generalizations of these two functions. The simplicity of this relationship enables us to easily translate the properties proved for one of them to the case of the other. We perform such conversions regarding distinguishability, decomposition techniques and positivity questions. Among others, we obtain the converse of Redei's theorem, generalization of the triple deletion property and expressions for these functions in some special cases.