Discrete Mathematics Seminar
Spring 2023


April 21  R. Griffin, URI  On gaps in the gaussian primes 
It is known that one cannot walk to infinity on the real line using only primes and steps of bounded length.
We will survey what is known on the same problem for Gaussian primes in
$\mathbb{Z}[i]$. The later problem is presently
unresolved. The MS presentation is based on papers by M. Das (Walking through the Gaussian Primes) and by
E. Gethner, S. Wagon, B. Wick (A Stroll through the Gaussian Primes).


April 14  M. Barrus, URI  Distinguishing chromatic numbers of hamiltonian circulant graphs 
The distinguishing chromatic number of a graph $G$ is the minimum number
of colors needed to properly color
the vertices so that no nontrivial symmetry of $G$ preserves the coloring.
Providing many examples of the twists and
turns symmetries introduce, we determine the distinguishing chromatic number of
circulant graphs with connection
set $\{ \pm 1, \pm k\}$. This is joint work with Jean Guillaume (Sacred Heart
University) and Benjamin Lantz (Wheaton College).


March 31  E. Barranca, URI  Characterizing some graphs which are recognizable by spectrum 
A graph is said to be determined by its spectrum (DS) if its adjacency
spectrum is not shared by any nonisomorphic
graphs. In this presentation we explore an extension to the question of
determining which graphs are DS. We say a graph
$G$ is recognizable by spectrum (RS) if any graph's spectrum determines if
it contains $G$ as an induced subgraph. We
will present examples of graphs which are RS and, in our attempt to characterize
small RS graphs, we will share results
focusing on trees and graphs with special cycles and cliques.

Fall 2022
December 2  D. Johnston, Trinity College, Hartford CT  Rainbow Turan Numbers and Rainbow Saturation 
Abstract:
Extremal graph theory asks questions of the form: What is the maximum or minimum size of a graph satisfying certain
conditions? For example, Turan investigated the maximum number of edges in a graph on $n$ vertices that avoids a given
subgraph $F$, denoted by $ex(n; F)$. Keevash, Mubayi, Sudakov and Verstraete extended this idea to graphs with proper
edgecolorings: colorings of the edges of a graph so that adjacent edges have different colors. For a given subgraph $F$, they
studied the maximum number of edges in a properly edgecolored graph on $n$ vertices that does not contain a rainbow copy
of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by $ex^*(n; F)$, is the
rainbow Turan number of $F$. In this talk, we examine rainbow Turan numbers and answer the question, In what way does
instead looking at the minimum number of edges in the context above make sense? by introducing the rainbow saturation
number of $F$. Host Mike Barrus.
