Georgia Institute of Technology College of Sciences

We discuss flows (and group-connectivity) in signed graphs, and prove a new result about group-connectivity in 3-edge-connected signed graphs. This is joint work with Alejandra Brewer Castano and Kathryn Nurse.

This talk will have two parts.  The first half will describe how to construct symplectic structures on trisected 4-manifolds. This construction is inspired by projective complex geometry and completely characterizes symplectic 4-manifolds among all smooth 4-manifolds.  The second half will address a curious phenomenon: symplectic 4-manifolds appear to not admit any interesting connected sum decompositions.  One potential explanation is that every embedded 3-sphere can be made contact-type.  I will outline some strategies to prove this from a trisections perspective, describe some of the obstructions, and give evidence that these obstructions may be overcome.

A class of graphs is said to be $\chi$-bounded with binding function $f$ if for every such graph $G$, it satisfies $\chi(G) \leq f(\omega(G)$, and polynomially $\chi$-bounded if $f$ is a polynomial. It was conjectured that chair-free graphs are perfectly divisible, and hence admit a quadratic $\chi$-binding function. In addition to confirming that chair-free graphs admit a quadratic $\chi$-binding function, we will extend the result by demonstrating that $t$-broom free graphs are polynomially $\chi$-bounded for any $t$ with binding function $f(\omega) = O(\omega^{t+1})$. A class of graphs is said to satisfy the Vizing bound if it admits the $\chi$-binding function $f(\omega) = \omega + 1$. It was conjectured that (fork, $K_3$)-free graphs would be 3-colorable, where fork is the graph obtained from $K_{1, 4}$ by subdividing two edges. This would also imply that (paw, fork)-free graphs satisfy the Vizing bound. We will prove this conjecture through a series of lemmas that constrain the structure of any minimal counterexample.