**James Oxley** (Keynote) — *Inductive tools for handling internally 4-connected binary matroids and graphs*

A matroid is 3-connected if it does not break up as a 1-sum or a 2-sum. Numerous problems for matroids reduce easily to the study of 3-connected matroids. Two powerful inductive tools for dealing with 3-connected matroids are Tutte's Wheels-and-Whirls Theorem and Seymour's Splitter Theorem. For several years, Carolyn Chun, Dillon Mayhew, and I have been seeking analogues of these theorems for internally 4-connected binary matroids, that is, binary matroids that do not break up as a 1-, 2-, or 3-sum. The class of such matroids includes the cycle matroids of internally 4-connected graphs, those 3-connected simple graphs that are 4-connected except for the possible presence of degree-3 vertices. This talk will report on our progress towards finding these analogues.

**Stan Dziobiak** — *Obstructions of apex classes of graphs*

The famous Graph Minor Theorem of Robertson and Seymour states that every minor-closed class ** C** of graphs can be characterized by a finite list of minor-minimal non-members, called

**Eric Gottlieb** — *Discrete fair division using posets*

Consider the problem of allocating a number of discrete goods among a set of individuals. If each individual ranks the items linearly according to their preferences, then an individual's ranking naturally induces a partial order *M* on sets of the items. *M* induces a partial order on the set of all allocations of items to individuals. *M* has been studied in different contexts by Stanley and others.

Each individual's partial order on allocations can then be viewed as a ballot in an election in which the allocations play the role of the candidates. In earlier work, some collaborators and I proposed a method for tabulating ballots when preferences are partially ordered. Cullinan, Hsiao, and Polett proposed a computationally simpler approach and proved that it has some nice properties. I will give some preliminary observations on the application of their approach to the fair division problem.

**Daniel Guillot** — *Coloring graphs drawn with no dependent crossings*

Král and Stacho showed that if a graph is drawn in the plane so that the vertices incident with crossed edges are all distinct, then the graph admits a 5-coloring. In the talk, I will discuss extensions of this result to graphs drawn on other surfaces.

**Ademir Hujdurovic** — *Pentavalent symmetric bicirculants*

**Bette Catherine Putnam** — *Bicircular Matroids with Circuits of Few Sizes*

**Jesse Taylor** — *On matroid minors that guarantee their duals as minors*

A fundamental operation for matroids is the construction of the dual. This talk solves the problem of determining all 3-connected binary matroids *N* such that, whenever a 3-connected binary matroid *M* has an *N*-minor, and *M* is large enough, *M* also has the dual of *N* as a minor. No previous familiarity with matroids will be assumed.

**Shaohui Wang** — *Multiplicative Zagreb indices of *k*-trees*

**Dong Ye** — *Perfect matchings and resonant patterns of fullerenes*

A fullerene is a 3-connected plane graph with only pentagonal faces and hexagonal faces. Let *F* be a fullerene graph. A perfect matching of *F* is a set *M* of disjoint edges such that every vertex of *F* is incident with exactly one edge in *M*. A resonant pattern of a fullerene graph *F* is a set **H** of disjoint hexagons whose deletion results in a subgraph with a perfect matching. In other words, *F* has a perfect matching *M* such that the edges of every hexagon in **H** alternate between *M* and *E*(*G*) \ *M*. In this talk, we will survey some old and new results in this direction.

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*Last modified September 14, 2018 *