The 1-Color Problem and the Brylawski Model

Abstract

In discrete tomography, the 1-color problem consists in determining the existence of a binary matrix with row and column sums equal to some given input values arranged in two vectors. These two vectors are said to be compatible if the associated 1-color problem has at least a solution. Here, we start from a vector of projections, and we define an algorithm to compute all the vectors compatible with it, then we show how to arrange them in a partial order structure, and we point out some of its combinatorial properties. Finally, we prove that this poset is a sublattice of the Brylawski lattice too, and we check some common properties.

On the Degree Sequences of Uniform Hypergraphs

“…The authors of [8], pointed out that these two conditions turn out to be sufficient in case of homogeneous horizontal and vertical projections, by showing their maximality w.r.t. the cardinality of the related sets of solutions.…”