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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 combinatorial 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 XG of a graph G was introduced by Stanley. In this paper we introduce a quasisymmetric generalization X 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 XG to χG(λ), the chromatic polynomial, we also… (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).