Let G be a finite graph with p vertices and x its chromatic polynomial. A combinatorial interpretation is given to the positive integer (-l)px(-A), where h is a positive integer, in terms of acyclic orientations of G. In particular, (-l)Px(-1) is the number of acyclic orientations of G. An application is given to the enumeration of labeled acyclic digraphs. An algebra of full binomial type, in ...

A unique sink orientation (USO) is an orientation of the n-dimensional cube graph (n-cube) such that every face (subcube) has a unique sink. The number of unique sink orientations is nΘ(2 n) [13]. If a cube orientation is not a USO, it contains a pseudo unique sink orientation (PUSO): an orientation of some subcube such that every proper face of it has a unique sink, but the subcube itself hasn...

Article history: Received 10 July 2006 Available online 29 August 2008

A classical result in graph theory asserts that every graph can be oriented so that the indegree and outdegree of each vertex differ by at most 1. We study the extent to which the result generalizes to uniform hypergraphs.

We find the asymptotic number of 2-orientations of quadrangulations with n inner faces, and of 3orientations of triangulations with n inner vertices. We also find the asymptotic number of prime 2-orientations (no separating quadrangle) and prime 3-orientations (no separating triangle). The estimates we find are of the form c ·n−αγn, for suitable constants c, α, γ, with α = 4 for 2-orientations ...