Bijective proofs of two broken circuit theorems

We prove, by means of explicit bijections, theorems of Whitney and Stanley that express the coefficients of the chromatic polynomial of a graph G and the number of acyclic orientations of G in terms of numbers of sets of edges that contain no broken circuits of G. Let G be a graph, and let a total ordering of its edge set E ( G ) be fixed. Words like “first” or “later,” when applied to edges, will always refer to this ordering. We adopt, unless the contrary is explicitly stated, the graph-theoretic notation and terminology of [l]; in particular, G is undirected and has no loops or multiple edges, p is the cardinality of the vertex set V(G) , andf(G, A ) is the chromatic polynomial. Following Whitney [3], we define a broken circuit to be a set of edges obtained by removing from some circuit in G its last edge. We shall give bijective proofs of the following two results. Whitney’s Theorem [3]. Let d , ( G ) be the collection of all sets A that consist of exactly i edges of G and contain no broken circuit; then f(G, A) = 2 (l)ildi(G)lAp-’