02:00pm  03:00pm 
BLOC 166 
Fu Liu
(University of California Davis)



+ A combinatorial analysis of Severi degrees



The classical Severi degree denoted by \$N^{d, \delta}\$ counts the number of curves of degree \$d\$ with \$\delta\$ nodes passing through \$\frac{d(d+3)}{2}  \delta\$ general points in the complex projective plane. Based on results by Brugallè and Mikhalkin, Fomin and Mikhalkin give formulas for computing \$N^{d, \delta}\$ using longedge graphs. Motivated by a conjecture of BlockColleyKennedy, we consider a special multivariate function associated to longedge graphs, and show that this function is always linear.
Applying the linearity result to FominMikhalkin's formula, we recover two known results for the classical Severi degrees. Further, in joint work with Osserman, the linearity result enables us to obtain a universal polynomiality property of Severi degrees on some families of toric surfaces. Since the proof of our linearity result is completely combinatorial, we are able to provide combinatorial formulas for the two unidentified power series appearing in the GöttscheYauZaslow formula.

03:10pm  04:10pm 

Contributed Session 1


BLOC 164 
Session A



03:10pm  03:30pm 
Katie Anders
(University of Texas at Tyler)


+ Odd behavior in the coefficients of reciprocals of binary power series


Let \$\mathcal{A}\$ be a finite subset of \$\mathbb{N}\$ including 0 and \$f_\mathcal{A}(n)\$ be the number of ways to write \$n = \sum_{i=0}^{\infty}\epsilon_i2^i\$, where \$\epsilon_i\in\mathcal{A}\$. The sequence \$\left(f_\mathcal{A}(n)\right) \bmod 2\$ is always periodic. Small examples suggest that the number of odd \$f_\mathcal{A}(n)\$'s in a period is at most 1 plus the number of even \$f_\mathcal{A}(n)\$'s in a period. We will discuss strong larger counterexamples and give four families of sets \$\left(\mathcal{A}_m\right)\$ with \$\left\mathcal{A}_m\right = 4\$ such that the proportion of odd \$f_{\mathcal{A}_m}(n)\$'s goes to 1 as \$m \to \infty\$.

03:30pm  03:50pm 
Christopher O'Neill
(Texas A&M University)


+ Shifting numerical monoids


Consider the family of numerical monoids \$S_n = \langle n, n + r_1, \ldots, n + r_k \rangle \subset \mathbb{N}\$ obtained by varying \$n\$. In this talk, we exhibit periodic behavior of the minimal presentations of \$S_n\$ when \$n\$ is sufficiently large. As a consequence, we characterize the eventual behavior of several arithmetic quantities arising in factorization theory. No background in factorization theory or numerical semigroups will be assumed for this talk.

03:50pm  04:10pm 
Suho Oh
(Texas State University)


+ A combinatorial problem coming from group theory


We consider a combinatorial problem(proposed by Thomas Keller), which originates from group theory. The task is to fill a grid with \$k\$ rows and infinitely many columns with integers according to certain rules.
We present the problem, sketch the tools needed for the proof, and propose a general version of the problem as a conjecture, which is closely related to generalized Hall's theorem for hypergraphs. This is a joint work with Eugene Curtin.



BLOC 148 
Session B



03:10pm  03:30pm 
Ismael Gonzalez Yero
(University of Cadiz, Spain)


+ Efficient open and efficient closed domination graphs


A graph is an efficient (open or closed, resp.) domination graph if there exists a subset of vertices whose (open or closed, resp.) neighborhoods partition its vertex set. Graphs that are efficient open domination graphs are investigated, as well as, graphs that are efficient open and efficient closed domination graphs at the same time. Several combinatorial and computational properties of efficient (open or closed, resp.) domination graphs are given. For instance, an NPcompleteness proof of deciding whether a given graph is an efficient (open and closed at the same) domination graph is presented. Moreover, several classes of graphs that are efficient open and/or closed domination graphs are discussed.

03:30pm  03:50pm 
LindsayKay Lauderdale
(University of Texas at Tyler)


+ Vertex minimal graphs with dihedral symmetry


Let \$D_{2n}\$ denote the dihedral group of order \$2n\$, where \$n\$ is an integer greater than three. In this article we build upon the findings of Haggard and McCarthy who, for certain values of \$n\$, each produced a vertex minimal graph whose automorphism group is isomorphic to \$D_{2n}\$. Specifically, Haggard considered the situation where \$\frac{n}{2}\$ or \$n\$ is a power of a prime number and McCarthy investigated the case when \$n\$ is not divisible by two, three nor five. Here we construct a vertex minimal graph whose automorphism group is isomorphic to \$D_{2n}\$ where \$n\$ is not divisible by four. These results provide a new geometric interpretation of the dihedral group.

03:50pm  04:10pm 
Cong Kang
(Texas A&M University at Galveston)


+ Metric dimension versus zero forcing number


The "metric dimension" \$dim(G)\$ of a graph \$G\$ is the minimum number of vertices such that every vertex of \$G\$ is uniquely determined by its vector of distances to the chosen vertices. For a graph \$G\$ with vertexes set \$V(G)\$, the "zero forcing number", \$Z(G)\$, of \$G\$ is the minimum cardinality of a set \$S\$ of black vertices (whereas vertices in \$V(G)S\$ are colored white) such that \$V(G)\$ is turned black after finitely many applications of `the colorchange rule': a white vertex is converted black if it is the only white neighbor of a black vertex. In this talk, we discuss the relationship between the metric dimension and the zero forcing number of graphs.
This talk is based on a joint work with Linda Eroh and Eunjeong Yi.


04:10pm  04:30pm 
BLOC 140 
Break

04:30pm  05:30pm 
BLOC 166 
Bruno Benedetti
(University of Miami)



+ Balinski's theorem and regularity of line arrangements



I will report on an ongoing project with Matteo Varbaro and Michela Di Marca (U Genova). Castelnuovo and Mumford introduced long ago a notion of regularity for ideals of polynomials. In combinatorics we also use of the word "regularity", for graphs in which all vertices have the same number of neighbors. We show an unexpected connection between the two notions, for Gorenstein arrangements of projective lines. If time permits, I will discuss extensions from arrangements of lines to arrangements of curves. The surprise here (joint with Barbara Bolognese, Matteo Varbaro) is that Balinski's theorem, "the graph of \$d\$polytopes is \$d\$connected", extends to this setup.

06:00pm  08:00pm 
BLOC 141 
Conference Dinner (catered)
