The Algebra, Combinatorics, Geometry, and Topology (ACGT) Seminar meets on Tuesdays at 1:00-1:50PM in Room 146 of the Adel Mathematics Building. If you are interested in giving a talk, please contact Dana C. Ernst, ACGT coordinator.

Note that talks are listed in reverse chronological order.

**Date:** November 26, December 3

**Speaker:** Bahattin Yildiz (NAU)

**Abstract:** A We will consider the homogeneous weights for several finite rings and explore the contribution of finite geometries into the construction of Gray maps for these homogeneous weights. We will try in the end to get a sense of what rings and weights would be appropriate to consider within the context of finite geometries.

**Date:** November 19

**Speaker:** Ye Chen (NAU)

**Abstract:** A graph has a k-locating-coloring if the vertices of the graph can be partitioned into k sets such that (1) the adjacent vertices are not in the same set and, (2) the distance coordinates vector which consists of the distances between the vertex and the k sets are different between any pair of vertices. The smallest k such that the graph G has a k-locating-coloring is the locating-chromatic number of G. I will first finish the proof of locating chromatic number of cycles, then move to the generalized Petersen graph.

**Date:** November 12

**Speaker:** Ye Chen (NAU)

**Abstract:** A graph has a k-locating-coloring if the vertices of the graph can be partitioned into k sets such that (1) the adjacent vertices are not in the same set and, (2) the distance coordinates vector which consists of the distances between the vertex and the k sets are different between any pair of vertices. The smallest k such that the graph G has a k-locating-coloring is the locating-chromatic number of G. I will go through some examples and results on paths and cycles in this seminar. Hopefully we can touch base on the locating chromatic number of generalized Petersen graph.

**Date:** October 29, November 5

**Speaker:** Dana C. Ernst (NAU)

**Abstract:** A Coxeter group can be thought of as a generalized reflection group, where the group is generated by a set of elements of order two and there are rules for how the generators interact with each other. Every element of a Coxeter group can be written as an expression in the generators, and if the number of generators in an expression is minimal, we say that the expression is reduced. Any two reduced expressions for the same group element are related by a sequence of commutations and so-called braid moves. We say that two reduced expressions are braid equivalent if they are related via a sequence of braid moves, and the corresponding equivalence classes are called braid classes. Each braid class can be encoded in terms of a braid graph, where each vertex is an element of the braid class and two vertices are connected by an edge whenever the corresponding reduced expressions are related via a single braid move. We will summarize recent progress concerning the taxonomy of braid graphs for Coxeter groups of types $A$ and $D$. Joint work with Fadi Awik, Jadyn Breland, Quentin Cadman, Jens Niemi, Jack Sullivan, and Jordan Wright.

**Date:** October 22

**Speaker:** Michael Falk (NAU)

**Abstract:** A simple graph determines a “graphic hyperplane arrangement,” the complement of which is a “partial configuration space,” whose fundamental group is a “GAG” (graphic arrangement group) or graphic pure braid group. This is a natural quotient of the pure braid group, with generators corresponding to edges of the graph and relations arising from 3- and 4-cliques. Not so much is known in general about these groups. We’ll state a theorem and present an example supporting a conjecture that GAGs are isomorphic to subgroups of direct products of pure braid groups. (Joint work with Dan Cohen.)

**Date:** October 15

**Speaker:** Alessandra Graf, University of Waterloo

**Abstract:** Let G be a graph and (V_1,…,V_m) be a vertex partition of G. An independent transversal (IT) of G with respect to (V_1,…,V_m) is an independent set {v_1,…,v_m} in G such that v_i is in V_i for each i in {1,…,m}. There exist various theorems that give sufficient conditions for the existence of ITs. These theorems have been used to solve problems in graph theory, hypergraphs, group theory, and theoretical computer science.
In this talk, we discuss a new poly-time algorithm for finding an IT under certain conditions and some applications of this algorithm to graph colorings and hypergraph matchings.

**Date:** October 8

**Speaker:** Michael Falk (NAU)

**Abstract:** One can associate to a graph G a natural quotient of the pure braid group, called a GAG (for graphic arrangement group). I’ll present a theorem and conjecture about injectivity of the natural map from a GAG to the product of the pure braid groups corresponding to the maximal cliques of the underlying graph, and then focus on establishing this property in one very interesting example.

**Date:** October 1

**Speaker:** Michael Falk (NAU)

**Abstract:** Graphic braid groups, part 3.
Abstract: One can associate to a graph G a natural (sub)quotient of the full braid group. In this episode we will examine the graphic braid group associated with the path of length two, and perhaps work through another example together, and then I will proceed to a discussion of GAGs and their properties.

**Date:** September 24

**Speaker:** Michael Falk (NAU)

**Abstract:** One can associate to a graph G a natural (sub)quotient of the full braid group. We will define this notion, and illustrate how the symmetry group of the isosceles right triangle tiling of the plane arises in this way. Last week I discussed the braid and pure braid groups; this week we’ll explore the GAG and graphic braid group associated with a graph, and work out an example together. I will also tell a share a few more stories and pictures from my sabbatical.

**Date:** September 17

**Speaker:** Michael Falk (NAU)

**Abstract:** One can associate to a graph G a natural (sub)quotient of the full braid group. We will define this notion, and illustrate how the symmetry group of the isosceles right triangle tiling of the plane arises in this way. I will also use part of the period to flesh out my sabbatical report from last Tuesday’s colloquium, with pictures and stories.

**Date:** September 3, 10

**Speaker:** Steve Wilson (NAU emeritus)

**Abstract:** This talk will touch on edge-colorings of graphs, particularly tetravalent graphs. We can think of these colorings as equivalence relations on the set of edges, or as functions from the set of edges to some fixed color set. We are particularly interested in ones whose symmetry group is large enough to be transitive on darts (i.e., directed edges) in the graph. We will show some of the variety of possibilities, motivated by their uses in BGCG constructions.