Noah Streib

Five-coloring graphs on the Klein bottle

We exhibit an explicit list of nine graphs such that a graph drawn in the Klein bottle is 5-colorable if and only if it has no subgraph isomorphic to a member of the list.

A Major League Baseball Team Uses Operations Research to Improve Draft Preparation

Preparing for the annual major league baseball draft is a difficult task; with 1,500 players selected each year, teams must evaluate and rank many hundreds of potential draftees. To evaluate the players, these teams send out scouts, baseball experts who make qualitative and quantitative observations and report their opinions to the team. However, scouts often disagree significantly in their opinions. We worked with a major league team to model and solve the problem of suggesting a consensus ranking of all players scouted by the teams representatives. Our methodology can also make in-season recommendations for dynamic scout scheduling based on the level of information each scout is likely to provide on each player, and the uncertainty in the “correct” overall ranking of each player. The team has been using the optimization tool we provided for the past two years, and a second major league team has also asked us to evaluate its ranking data.

The total linear discrepancy of an ordered set

In this paper we introduce the notion of the total linear discrepancy of a poset as a way of measuring the fairness of linear extensions. If L is a linear extension of a poset P, and x,y is an incomparable pair in P, the height difference between x and y in L is |L(x)-L(y)|. The total linear discrepancy of P in L is the sum over all incomparable pairs of these height differences. The total linear discrepancy of P is the minimum of this sum taken over all linear extensions L of P. While the problem of computing the (ordinary) linear discrepancy of a poset is NP-complete, the total linear discrepancy can be computed in polynomial time. Indeed, in this paper, we characterize those linear extensions that are optimal for total linear discrepancy. The characterization provides an easy way to count the number of optimal linear extensions.

Pebbling Graphs of Diameter Three and Four

Abstract Given a configuration of pebbles on the vertices of a connected graph G, a pebbling move is defined as the removal of two pebbles from some vertex, and the placement of one of these on an adjacent vertex. The pebbling number of a graph G is the smallest integer k such that for each vertex v and each configuration of k pebbles on G there is a sequence of pebbling moves that places at least one pebble on v. We improve on the bound of Bukh by showing that the pebbling number of a graph of diameter three on n vertices is at most ⌊ 3 n / 2 ⌋ + 2 , and this bound is best possible. We obtain an asymptotically best possible bound of 3 n / 2 + Θ ( 1 ) for the pebbling number of graphs of diameter four. Finally, we prove an asymptotic upper bound for the pebbling number of graphs of diameter d, namely ( 2 ⌈ d 2 ⌉ − 1 ) n + O ( 1 ) , and this also improves a bound given by Bukh.

Hamiltonian Cycles and Symmetric Chains in Boolean Lattices

Let B(n) be the subset lattice of

Dimension and Height for Posets with Planar Cover Graphs

{\{1,2,\dots, n\}.}

Dimension Preserving Contractions and a Finite List of 3-Irreducible Posets

{1,2,⋯,n}. Sperner’s theorem states that the width of B(n) is equal to the size of its biggest level. There have been several elegant proofs of this result, including an approach that shows that B(n) has a symmetric chain partition. Another famous result concerning B(n) is that its cover graph is hamiltonian. Motivated by these ideas and by the Middle Two Levels conjecture, we consider posets that have the Hamiltonian Cycle–Symmetric Chain Partition (HC-SCP) property. A poset of width w has this property if its cover graph has a hamiltonian cycle which parses into w symmetric chains. We show that the subset lattices have the HC-SCP property, and we obtain this result as a special case of a more general treatment.

Dimension and height for posets with planar cover graphs

For positive integers w and k, two vectors A and B from Z w are called k-crossing if there are two coordinates i and j such that A i - B i ? k and B j - A j ? k . What is the maximum size of a family of pairwise 1-crossing and pairwise non-k-crossing vectors in Z w ? We state a conjecture that the answer is k w - 1 . We prove the conjecture for w ? 3 and provide weaker upper bounds for w ? 4 . Also, for all k and w, we construct several quite different examples of families of desired size k w - 1 . This research is motivated by a natural question concerning the width of the lattice of maximum antichains of a partially ordered set.