Talks concerning the study of embeddings of graphs in the 3-dimensional space or surfaces and graphs as topological spaces. Organizers: Thomas Mattman and Andrei Pavelescu.
Topological graph theory studies how graphs can be embedded in the 3-dimensional space and on surfaces, focusing on the relationships between graph properties and the topology of the surfaces in which they are embedded.
Organizers: Thomas Mattman and Andrei Pavelescu
Suppose you have a subset $S$ of the vertices of a planar graph which contains at least one vertex from every face. Then $S$ must have at least half of the vertices, and for some planar graphs *every* such $S$ must have at least half of the vertices. We believe this extends to higher dimensions, but don’t really know why, and have found some situational evidence (but also some counter-evidence). This is based on joint work with Michael Dobbins and Seunghun Lee.
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We say that a simplicial $n$-complex is intrinsically linked if every embedding of its polyhedron into $(2n+1)$-dimensional Euclidean space contains a pair of disjoint $n$-spheres with a nonzero linking number. Several examples of intrinsically linked $n$-complexes are known. In this talk, we present a new example of such an $n$-complex.
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A $2$-cycle on a graph $G=(V,E)$ is a function $d: E\times E\to \mathbb{Z}$ such that for each edge $e$, both $d(e, \cdot)$ and $d(\cdot, e)$ are circulations on $G$. For an oriented cycle $C$ and an edge $e$ of $G$, define $C(e)=+1$ if $C$ traverses $e$ in forward direction, and $C(e)=-1$ if $C$ traverses $e$ in backward direction. Then examples of $2$-cycles are: take two vertex-disjoint oriented cycles $C$ and $D$ of $G$ and define $d(e,f) = C(e)D(f)$. Also on each $K_{3,3}$- and $K_5$-subdivision are $2$-cycles. In this talk, we show that each $2$-cycle on $G$ can be written as a sum of four types of special $2$-cycles. This is joint work with Serguei Norine and Robin Thomas.
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Decomposing the complete graph into arbitrary graph is a challenging and di cult problem in graph theory. In in this paper, we prove that the complete graph K4m+1 can be decomposed into 4m + 1 copies of an arbitrary tree with m edges and m copies of a Hamiltonian cycle whenever 4m+1 is a prime.
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Determining how to build a minimal genus embedding of a graph is a classical and frequently challenging problem in topological graph theory. Here, we will be interested in Cartesian products of graphs where their fiber structures and symmetries can sometimes be leveraged to efficiently build embeddings and determine the genera of such graphs. More specifically, we will discuss work done towards a classification of all Cartesian products of graphs that embed on the torus where we leverage basic tools from combinatorial topology and determine the genera of certain graphs along the way. This work was part of an undergraduate summer research project with Beppy Badgett, and time permitting, we will briefly discuss ideas for future projects in this area that only requires some background in undergraduate graph theory and surface topology to get started.
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An $(n,k)$-flower $F(n,k)$ is the shadow of the closure of an $n$-braid $(\sigma_{1}\sigma_{2}\cdots\sigma_{n-1})^{k}$.\\ C. Lamm and V. O. Manturov independently showed the following: Let $K$ be a knot and $n\geq \mathrm{braid}(K)$. Then $K$ has $F(n,k)$ as its shadow for some $k$. We show the following: Let $K$ be a knot and $k\geq \mathrm{bridge}(K)$. Then $K$ has $F(n,k)$ as its shadow for some $n$. As a corollary, we show $\mathrm{bridge}(K)=\mathrm{lr}(K)$ where $\mathrm{lr}(K)$ is the left-right number of $K$. This gives us a new definition of the bridge number of a knot.
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A simplicial complex is said to be critical (or forbidden) for the 3-sphere $S^3$ if it cannot be embedded in $S^3$, but becomes embeddable upon removing the open star of any simplex in its second barycentric subdivision. We classify all critical complexes for $S^3$ that decompose as $(G \times S^1) \cup H$, where $G$ and $H$ are graphs whose intersection $G \cap H$ consists solely of vertices of $H$. This is a joint work with Mario Eudave-Munoz.
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Let $H$ be a given graph. A graph $G$ is $H$-linked if, for any injective map $\phi: V(H)\to V(G)$, $G$ contains a subdivision of $H$ rooted at the images of $V(H)$. A classic result of Seymour and Thomassen shows that every 4-connected plane triangulation is $K_2$-linked. Ellingham, Plummer and Yu proved that every 4-connected plane triangulation is $K_4^-$-linked. However, not all 4-connected surface triangulation is $K_4^-$-linked. In this talk, we focus on some recent developments on graph linkages on surfaces. This is based on joint work with Moser, Stephens, and Zha.
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The Graph Minor Theorem of Robertson and Seymour implies a finite set of obstructions for any minor closed graph property. We show that there are only three obstructions to knotless embedding of size 23, which is far fewer than the 92 of size 22 and the hundreds known to exist at larger sizes. We describe several other topological properties whose obstruction set demonstrates a similar dip at small size. For order ten graphs, we classify the 35 obstructions to knotless embedding and the 49 maximal knotless graphs. This work is collaborated with Thomas Mattman.
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For any d > 0, define $G(\mathbb{Q}^n, d)$ to be the graph whose vertices are points of the rational space $\mathbb{Q}^n$ with any two vertices being adjacent if and only if they are a Euclidean distance $d$ apart. Such a graph is only of interest if $d$ is a distance actually realized between points of $\mathbb{Q}^n$, so we might as well assume that is the case. In this talk, we will ask for which $n$ and distances $d_1, d_2$ the graphs $G(\mathbb{Q}^n, d_1)$ and $G(\mathbb{Q}^n, d_2)$ are isomorphic. A resolution will be given for $n \leq 4$, and we will then present, by way of drawing a bunch of pictures, a method that, perhaps with some ingenuity, could be extended to answer this question for general $n$.
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A handcuff graph is a graph consisting of disjoint two loops and connected by an edge. The lattice stick number of a handcuff graph is the minimum number of sticks required to embed the graph in a lattice space. Previous studies have shown that the lattice stick numbers of the trivial handcuff graph and the Hopf-linked handcuff graph are 9 and 11, respectively, and these are the only graphs whose lattice stick number is 13 or less. In this talk, we utilize a squeezing method to identify all models with a lattice stick number of at most 14.
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In joint work with Louis Kauffman and Susan Williams, we give an elementary introduction to the Penrose polynomial for cubic graphs and explore the combinatorial significance of its coefficients. No special background is assumed.
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Since Kauffman first introduced the concept of ribbonlength for knots and links, many researchers, including Denne, have studied ribbonlength using folded ribbons. In several studies, upper bounds for ribbonlength were obtained by folding the ribbon in the shape of a right isosceles triangle. In this presentation, we introduce a new method of folding the ribbon into an equilateral triangle shape instead of the right isosceles triangle. We will then relate this method to three-page presentations, which are variations of the arc presentation. In this process, we present new upper bounds for the ribbonlength of the Hopf link and the trefoil knot.
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The intrinsic properties of carbon nanosheets derived from their basic molecular structure through a self-similar pattern have attracted much interest among researchers. Graph-theoretical methods are used to identify certain molecular descriptors known as topological indices, which are highly useful in connecting molecules to their physical attributes. Several chemical characteristics have been correlated with degree and neighborhood degree sum-based topological indices, which have been investigated extensively. Our current research establishes the use of topological indices in studying the newly synthesized carbon allotropes $\delta$-graphene, $\delta$-graphyne and $\delta$-graphdiyne. Furthermore, the complexity and information of carbon allotropes can be discussed in terms of Generalized Fractal Dimensions $(\mathrm{GFD})$, which are newly constructed based on Renyi entropy using some types of topological indices. The study of GFD indices of graphs is gaining importance as a measure of the complexity of basic coupling and as a tool for characterizing structural properties. We have estimated various topological indices, including graph-based GFD values of these structures obtained using the GFD method based on Renyi entropy.
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In 1930, Ramsey proved that for every positive integer $r$, every sufficiently large graph contains as an induced subgraph either $K_r$ or an independent set of size $r$. In this talk, I will give analogous characterizations for increasing levels of connectivity. This presentation combines work from two projects: the first with Guoli Ding and Bogdan Oporowski, and the second with Mark Ellingham.
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