Georgia Institute of Technology College of Sciences

A Roman dominating function of a graph G is a function f which maps V(G) to {0, 1, 2} such that whenever f(v)=0, there exists a vertex u adjacent to v such that f(u)=2. The weight of f is w(f) = \sum_{v \in V(G)} f(v). The Roman domination number \gamma_R(G) of G is the minimum weight of a Roman dominating function of G. Chambers, Kinnersley, Prince and West conjectured that \gamma_R(G) is at most the ceiling 2n/3 for any 2-connected graph G of n vertices. In this talk, we will give counter-examples to the conjecture, and proves that \gamma_R(G) is at most the maximum among the ceiling of 2n/3 and 23n/34 for any 2-connected graph G of n vertices. This is joint work with Gerard Jennhwa Chang.

Steinberg's Conjecture states that any planar graph without cycles of length four or five is three colorable. Borodin, Glebov, Montassier, and Raspaud showed that planar graphs without cycles of length four, five, or seven are three colorable and Borodin and Glebov showed that planar graphs without five cycles or triangles at distance at most two apart are three colorable. We prove a statement similar to both of these results: that any planar graph with no cycles of length four through six or cycles of length seven with incident triangles distance exactly two apart are three colorable. Special thanks to Robin Thomas for substantial contributions in the development of the proof.

The property that a graph has an embedding in projective plane is closed under taking minors. So by the well known theorem of Robertson and Seymour, there exists a finite list of minor-minimal graphs, call it L, such that a given graph G is projective planar if and only if G does not contain any graph isomorphic to a member of L as a minor. Glover, Huneke and Wang found 35 graphs in L, and Archdeacon proved that those are all the members of L. In this talk we show a new strategy for finding the list L. Our approach is based on conditioning on the connectivity of a member of L. Assume G is a member of L. If G is not 3-connected then the structure of G is well understood. In the case that G is 3-connected, the problem breaks down into two main cases, either G has an internal separation of order three or G is internally 4-connected . In this talk we find the set of all 3-connected minor minimal non-projective planar graphs with an internal 3-separation. This is joint work with Luke Postle and Robin Thomas.

First-Fit is an online algorithm that partitions the elements of a poset into chains. When presented with a new element x, First-Fit adds x to the first chain whose elements are all comparable to x. In 2004, Pemmaraju, Raman, and Varadarajan introduced the Column Construction Method to prove that when P is an interval order of width w, First-Fit partitions P into at most 10w chains. This bound was subsequently improved to 8w by Brightwell, Kierstead, and Trotter, and independently by Narayanaswamy and Babu. The poset r+s is the disjoint union of a chain of size r and a chain of size s. A poset is an interval order if and only if it does not contain 2+2 as an induced subposet. Bosek, Krawczyk, and Szczypka proved that if P is an (r+r)-free poset of width w, then First-Fit partitions P into at most 3rw^2 chains and asked whether the bound can be improved from O(w^2) to O(w). We answer this question in the affirmative. By generalizing the Column Construction Method, we show that if P is an (r+s)-free poset of width w, then First-Fit partitions P into at most 8(r-1)(s-1)w chains. This is joint work with Gwena\"el Joret.