A Partition of Connected Graphs

Abstract

We define an algorithm $k$ which takes a connected graph $G$ on a totally ordered vertex set and returns an increasing tree $R$ (which is not necessarily a subtree of $G$). We characterize the set of graphs $G$ such that $k(G)=R$. Because this set has a simple structure (it is isomorphic to a product of non-empty power sets), it is easy to evaluate certain graph invariants in terms of increasing trees. In particular, we prove that, up to sign, the coefficient of $x^q$ in the chromatic polynomial $\chi_G(x)$ is the number of increasing forests with $q$ components that satisfy a condition that we call $G$-connectedness. We also find a bijection between increasing $G$-connected trees and broken circuit free subtrees of $G$.