Bijective proofs of proper coloring theorems

TL;DR

This paper provides bijective proofs for key theorems relating the chromatic polynomial and symmetric function, unifying multiple expressions through only two core bijections, enhancing combinatorial understanding.

Contribution

It establishes all six classical expressions of chromatic invariants bijectively using only two bijections, simplifying and unifying previous proofs.

Findings

All six expressions are bijectively proven.

The same bijections apply to both polynomial and symmetric function cases.

The proof simplifies understanding of graph invariants.

Abstract

The chromatic polynomial and its generalization, the chromatic symmetric function, are two important graph invariants. Celebrated theorems of Birkhoff, Whitney, and Stanley show how both objects can be expressed in three different ways: as sums over all spanning subgraphs, as sums over spanning subgraphs with no broken circuits, and in terms of acyclic orientations with compatible colorings. We establish all six of these expressions bijectively. In fact, we do this with only two bijections, as the proofs in the symmetric function setting are obtained using the same bijections as in the polynomial case and the bijection for broken circuits is just a restriction of the one for all spanning subgraphs.