Some Extremal Problems on Graphs and Partial Orders
A unichain in a product poset P x Q is a chain in which the value of one coordinate is fixed. A semiantichain in P x Q is a family S such that (u, v) < ( u', v') for two elements of S only if u < u' and v < v' . Saks and West conjectured that for every product of partial orders, the maximum size of a semiantichain equals the minimum number of unichains needed to cover the product. We prove the case where both factors have width 2. We also use the characterization of product graphs that are perfect to prove other special cases, including the case where both factors have height 2. Finally, we make an observation about the case where both factors have dimension 2.
University of Illinois - Urbana-Champaign
Networked Digital Library of Theses and Dissertations
University of Illinois at Urbana-Champaign