Saturday, November 21, 2020

The Cascadia Combinatorial Feast (formerly known as the Combinatorial Potlatch) is an irregularly scheduled, floating, one-day conference. It has been held for many years at various locations around Puget Sound and southern British Columbia, and is an opportunity for combinatorialists in the region to gather informally for a day of invited talks and conversation. While most who attend work in, or near, the Puget Sound basin, all are welcome.  Typically there are three talks given by speakers who are visiting or new to the area, along with breaks for coffee and lunch. Many participants remain for dinner at a local restaurant or pub.

This year, we've decided to change the name of the conference. The original name, Potlatch, referred to a ceremonial feast among certain First Nations of the northwest Pacific coast.

This fall's conference will be hosted on Zoom by the Department of Mathematics and Statistics at the University of Victoria on Saturday, November 21, 2020.

The password has been sent to everyone on our mailing list, and can also be obtained by emailing Amites Sarkar at the address below.

### Schedule

•   9:45 AM Mike Henning
• 10:35 AM Coffee break
• 11:00 AM Natasha Morrison
• 11:50 AM Lunch and informal conversation
•   1:00 PM Chip Klostermeyer
•   1:50 PM Closing remarks
•   2:00 PM Networking and socializing (details TBA)
•   3:00 PM End of conference

### Upper Bounds on the Domination and Total Domination Numbers of a Graph in terms of Minimum Degree

In this talk, we present a survey of the currently best known upper bounds on the domination number $\gamma(G)$ and total domination number $\gamma_t(G)$ of a graph $G$ in terms of its order $n$ and minimum degree $\delta$. We focus mainly on results for small values of $\delta$, namely $\delta \le 6$. We present a sketch proof, that combines an algorithmic approach using vertex weighting arguments together with discharging methods, of the result that if $\delta = 6$, then $\gamma(G) \le \frac{127}{418}\, n \approx 0.30382775 \, n$. We also present a sketch proof, that uses transversals in hypergraphs, of the result that if $\delta = 6$, then $\gamma_t(G) \le \frac{4}{11}\, n \approx 0.363636 \, n$.

### The Typical Structure of Sets with Small Sumset

One of the central objects of interest in additive combinatorics is the sumset $A+B= \{a+b:a \in A, b \in B\}$ of two sets $A,B$ of integers.

Our main theorem, which improves results of Green and Morris, and of Mazur, implies that the following holds for every fixed $\lambda > 2$ and every $k>(\log n)^4$: if $\omega$ goes to infinity as $n$ goes to infinity (arbitrarily slowly), then almost all sets $A$ of $[n]$ with $|A| = k$ and $|A + A| < \lambda k$ are contained in an arithmetic progression of length $\lambda k/2 + \omega$.

This is joint work with Marcelo Campos, Mauricio Collares, Rob Morris and Victor Souza.

### Eternal Chromatic Numbers of Graphs

We study two eternal, or dynamic, variants of graph coloring. In the first version, starting with a proper coloring of graph $G$, an infinite sequence of requests occurs, each of which forces a vertex to change colors while maintaining a proper coloring. The minimum number of colors needed for $G$ over any such sequence is the eternal chromatic number of $G$. Bounds on the parameter are given.

The second problem we consider is a version of game coloring. Recall that game coloring is a two-player game in which each player properly colors one vertex of a graph at a time until all the vertices are colored. In the eternal version of game coloring, the vertices are colored and then subsequently re-colored from a color set over an infinite sequence of rounds. In a given round, each vertex is colored, or re-colored, once, so that a proper coloring is maintained. Player 1 wants to maintain a proper coloring forever, while Player 2 wants to force the coloring process to fail. The eternal game chromatic number of a graph $G$ is the minimum number of colors needed in the color set so that Player 1 can always win the game on $G$. Bounds are shown on some elementary classes of graphs. A number of conjectures and questions are posed for each version of the problem.

### Organizers

• Nancy Ann Neudauer, Pacific University, nancy (at) pacificu (dot) edu, Program Chair
• Amites Sarkar, Western Washington University, amites (dot) sarkar (at) wwu (dot) edu, Communications Chair
• Gary MacGillivray, University of Victoria, gmacgill (a) uvic (dot) ca, Local Arrangements Chair
Last updated: November 13, 2020, http://faculty.wwu.edu/sarkara/potlatch2020.html