Contact Russ Woodroofe ( russ.woodroofeVertex@famnit.upr.si ) to tell us you're interested in speaking and/or attending.

### Participants giving talks

### Participants not giving talks

### Abstracts

- Ron Gould (keynote speaker) / abstract
- Sarah Allred / abstract
- Rachel Barber / abstract
- Neal Bushaw / abstract
- Tara Fife / abstract
- Tiansi Li / abstract
- Jianbing Liu / abstract
- Andrei Pavelescu / abstract
- Josephine Reynes / abstract
- Bernd Schroeder / abstract
- Vaidy Sivaraman / abstract
- Semin Yoo / abstract

- Ted Dobson
- Laura Sheppardson
- Russ Woodroofe

The changing face of graph saturation

Given a graph *H*, we say that a graph *G* is * H-saturated* if it does not
contain

Since the introduction of saturation numbers, the topic of graph saturation has seen a variety of extensions and modifications. In this talk I will address a few of these ideas.

In particular, the *saturation spectrum* of the family of *H*-saturated
graphs on *n* vertices is the set of all possible sizes (|*E*(*G*)|) of an
*H*-saturated graph on *n* vertices. We look at a number of recent results
that determine the saturation spectrum for a variety of graphs.

For a graph *H*, a graph *G* is *weakly H-saturated* if

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CL-Shellable posets with no EL-shellings

Lexicographic shellability is defined and studied by Björner and Wachs as a way to compute the homology groups of shellable posets by either labeling edges (for EL-shellable posets) or labeling pairs of edges and maximal chains (for CL-shellable posets) in the Hasse diagrams. In this talk, I will first introduce the definition of EL-shellability and CL-shellability along with some examples, and how one can compute the homology groups of an EL-shellable poset given an EL-shelling. Then I will present two examples of CL-shellable posets that are not EL-shellable, which answers the question posed by Björner and Wachs that EL-shellability and CL-shellability are not equivalent for both graded and ungraded posets.

On weighted modulo orientations of graphs

The modulo orientation problem seeks an orientation, called modulo *k*-orientation with an odd *k*, of an undirected graph such that the indegree is congruent to outdegree modulo *k*.
Jaeger conjectured that every (2*k*-2)-edge-connected graph has a modulo *k*-orientation.
In connection to the 5-flow conjecture, we studied the problem of modulo
5-orientation for given multigraphic degree sequences.
We also showed that for any (*k*+1)-edge-connected graph *G* with a bounded matching number
has a modulo *k*-orientation with essentially finitely many exceptions.
In 2018, Esperet, De Verclos, Le and Thomass introduced the problem that for an odd prime *p*, whether there exists an orientation *D* of a graph *G* for any mapping *f*: *E*(*G*) → ℤ_{p}^{*} and any ℤ_{p}-boundary *b* of *G*, such that under *D*, at every vertex, the net out *f*-flow is the same as *b*(*v*) in ℤ_{p}. Such an orientation *D* is called an (*f*, *b*; *p*)-orientation of *G*. Esperet et al indicated that this problem is closely related to modulo *p*-orientations of graphs, including Tutte’s nowhere zero 3-flow conjecture. Utilizing properties of additive bases and contractible configurations, we showed that every (4*p*-6)-edge-connected planar graph *G* admits an (*f*, *b*; *p*)-orientation and that every (12*p*^{2} - 28*p *+ 15)-edge-connected signed graphs admit (*f*, *b*; *p*)-orientations. We also reduced the Esperet et al’s edge-connectivity lower bound for certain graphs families including complete graphs, chordal graphs and bipartite graphs.

Topological properties of maximal linklessly embeddable graphs

A simple graph is called intrinsically linked if every embedding of it into *R*^{3} contains a non-trivial link. Campbell et al. showed that a graph on *n* ≥ 6 vertices and at least 4*n *- 9 edges is intrinsically linked because it contains a *K*_{6}; this sets the upper bound for the size of a maximal linklessly embeddable graph on *n* vertices at 4*n *- 10. No results about a reasonable lower bound are known. In this talk, I will present some results on the structure of maximal linklessly embeddable graphs and present some candidates for the lower bound. I’ll also discuss applications to the simultaneous (non)linking of a graph and its complement.

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The automorphism conjecture for ordered sets

The automorphism conjecture for ordered sets states that the ratio of the number of automorphisms to the number of endomorphisms goes to zero as the size of the underlying ordered set goes to infinity. This talk will outline a method to determine, for a given width, the indecomposable ordered sets for which the number of automorphisms is maximal. This result can be used to prove the automorphism conjecture for ordered sets of width less than or equal to 10. Perhaps more importantly, the requisite insights into the structure of automorphisms of ordered sets could be the key to resolving the automorphism conjecture in general.

Chi-boundedness: a tale of two invariants

The chromatic number and the clique number are two fundamental invariants associated with a graph. Graphs in which these two numbers are the same for every induced subgraph, called perfect graphs, have rich structure, and fantastic properties, resulting in a well-developed theory, both structurally and algorithmically. András Gyárfás proposed to relax this concept to the notion of a "chi-bounded" family of graphs. I will survey some recent developments on chi-boundedness, and conclude with some fascinating open problems.

The dot-binomial coefficients

The Gaussian binomial coefficient (*n**k*)_{q} (or the *q*-binomial coefficient) is a polynomial in *q*, where *q* is a prime power. It can be described combinatorial in several ways. For example, it counts the number of *k*-dimensional subspaces of |F^{n}_{q} over |F_{q}. In this talk, I will discuss a new binomial coefficient, called the *dot-binomial coefficient* (*n**k*)_{d} which counts the number of *k*-dimensional quadratic subspaces of Euclidean type in (|F^{n}_{q}, *x*_{1}^{2} + ⋅⋅⋅ + *x*_{n}^{2}). We will also compare it with the Gaussian binomial coefficient.