Hypercube orientations with only two in-degrees

Hypercube orientations with only two in-degrees

We consider the problem of orienting the edges of the n-dimensional hypercube so only two different in-degrees a and b occur. We show that this can be done, for two specified in-degrees, if and only if an obvious necessary condition holds. Namely, there exist non-negative integers s and t so that s + t = 2n and as + bt = n2. This is connected to a question arising from constructing a strategy f...

Only knowing with degrees of confidence

A new logic of belief (in the “Only knowing” family) with confidence levels is presented. The logic allows a natural distinction between explicit and implicit belief representations, where the explicit form directly expresses its models. The explicit form can be found by applying a set of equivalence preserving rewriting rules to the implicit form. The rewriting process is performed entirely wi...

Orientations of Graphs with Prescribed Weighted Out-Degrees

If we want to apply Galvin’s kernel method to show that a graph G satisfies a certain coloring property, we have to find an appropriate orientation of G. This motivated us to investigate the complexity of the following orientation problem. The input is a graph G and two vertex functions f, g : V (G) → N. Then the question is whether there exists an orientation D of G such that each vertex v ∈ V...

On the Number of Planar Orientations with Prescribed Degrees

We deal with the asymptotic enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings, and ice models. The notion of orientations with outdegrees prescribed by a function α : V → N unifies many different combinatorial structures, including the afore mentioned. We call these orientations α-orien...

On Finite Groups With Two Irreducible Character Degrees