Graph-like Scheduling Problems and Property B

Breuer and Klivans defined a diverse class of scheduling problems in terms of Boolean formulas with atomic clauses that are inequalities. We consider what we call graph-like scheduling problems. These are Boolean formulas that are conjunctions of disjunctions of atomic clauses $(x_i \neq x_j)$. These problems generalize proper coloring in graphs and hypergraphs. We focus on the existence of a solution with all $x_i$ taking the value of $0$ or $1$ (i.e. problems analogous to the bipartite case). When a graph-like scheduling problem has such a solution, we say it has property B just as is done for $2$-colorable hypergraphs. We define the notion of a $\lambda$-uniform graph-like scheduling problem for any integer partition $\lambda$. Some bounds are attained for the size of the smallest $\lambda$-uniform graph-like scheduling problems without property B. We make use of both random and constructive methods to obtain bounds. Just as in the case of hypergraphs finding tight bounds remains an open problem

Boundary measurement and sign variation in real projective space

We define two generalizations of the totally nonnegative Grassmannian and determine their topology in the case of real projective space. We find the spaces to be PL manifolds with boundary which are homotopy equivalent to another real projective space of smaller dimension. One generalization makes use of sign variation while the other uses boundary measurement. Spaces arising from boundary measurement are shown to admit Cohen-Macaulay triangulations.Comment: v2 contains corrections of minor typos and is the accepted version to appear in Ann. Inst. Henri Poincar\'e

Scheduling Problems and Generalized Graph Coloring

International audienceWe define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, andsimplicial complexes. To this coloring there is an associated symmetric function in noncommuting variables for whichwe give a deletion-contraction formula. In the case of graphs our symmetric function in noncommuting variablesagrees with the chromatic symmetric function in noncommuting variables of Gebhard and Sagan. Our vertex coloringis a special case of the scheduling problems defined by Breuer and Klivans. We show how the deletion-contractionlaw can be applied to scheduling problems