Hopf Algebra Methods in Graph Theory

In this paper we introduce a Hopf algebraic framework for studying invariants of graphs, matroids, and other combinatorial structures. We begin by defining a category of objects, called Whitney systems, which are set systems having the minimum amount of structure necessary in order to have a sensible notion of connected subset, and which generalize graphs and matroids in several ways. Associated to any family P of Whitney systems, which is closed under restrictions and disjoint unions, is a certain Hopf algebra, generalizing the multiplicative formal group law, whose dual is isomorphic to the algebra of invariants defined on P , and whose continuous dual is isomorphic to an important subalgebra of invariants, called restriction invariants on P . We prove a structure theorem, namely, that the continuous dual of such a Hopf algebra is isomorphic to the polynomial algebra having the set of connected isomorphism types in P as indeterminates. We also introduce a general transformation, essentially the transpose of the logarithm map, which maps the class of restriction invariants onto the set of additive invariants on P . In a fundamental paper ([13]), H. Whitney showed that the chromatic polynomial of a graph G could be determined by examining only the doubly connected subgraphs of G; thus, in principle, it could be computed using less information than required by previously known techniques (see [4] and [12]). This result was derived independently in [10] by W. Tutte, who gave applications to the reconstruction problem in [11]. Some extensions were given by P. Erdös, L. Lovász and J. Spencer in [5]. Considerable clarification of Whitney’s work was given in [1] by N. Biggs, who also discussed its applications to statistical mechanics in [2] and [3]. In the second half of his paper [13] Whitney introduced a transformation from a set of graph invariants {mij}, which determine the chromatic polynomial and satisfy a kind of multiplicative property, to a set of invariants {fij}, which are additive (see section 6 of this paper). This transformation is invertible, thus the study of chromatic polynomials reduces to the study of the additive invariants fij. The program which he then set forth was to attack the four-color problem by systematically studying inequalities involving the fij. Unfortunately, the proofs in his paper were very obscure and, in particular, he gave no explicit technique for computing the mij or fij in terms of doubly connected subgraphs. Consequently, one is left without any clear approach to the problem of investigating the relations satisfied by the fij.