The graph algebra is a commutative, cocommutative, graded, connected incidence Hopf algebra, whose basis elements correspond to finite graphs and whose Hopf product and coproduct admit simple com-binatorial descriptions. We give a new formula for the antipode in the graph algebra in terms of acyclic orientations; our formula contains many fewer terms than… (More)
The chromatic symmetric function X G of a graph G was introduced by Stan-ley. In this paper we introduce a quasisymmetric generalization X k G called the k-chromatic quasisymmetric function of G and show that it is positive in the fundamental basis for the quasisymmetric functions. Following the specialization of X G to χ G (λ), the chromatic polynomial, we… (More)
As an example of a logic circuit design problem, we consider the two-inverter puzzle  which may be stated as follows:Using any number of AND, OR gates but no more than two NOT gates (inverters), design a logic circuit with three inputs i1, i2, i3 and three outputs o1, o2, 03 such that o1=NOT (il), o2=NOT (i2), o3=NOT (i3).