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.

摘要

色多项式及其推广，色对称函数，是两个重要的图不变量。Birkhoff、Whitney 和 Stanley 的著名定理展示了这两个对象如何以三种不同的方式表示：作为所有跨越子图的总和，作为没有断路的跨越子图的总和，以及根据具有兼容着色的非循环方向。我们双射地建立所有这六个表达式。事实上，我们只用两个双射来做到这一点，因为对称函数设置中的证明是使用与多项式情况相同的双射获得的，并且断路的双射只是对所有生成子图的双射的限制。