Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

The chromatic index $\chi'(G)$ of a graph $G$ is the least number of colors assigned to the edges of $G$ such that no two adjacent edges receive the same color. The total chromatic number $\chi''(G)$ of a graph $G$ is the least number of colors assigned to the edges and vertices of $G$ such that no two adjacent edges receive the same color, no two adjacent vertices receive the same color and no edge has the same color as its two endpoints. The chromatic index and the total chromatic number are two of few fundamental graph parameters, and their correlation has always been a subject of intensive study in graph theory.

By definition, $\chi'(G) \le \chi''(G)$ for every graph $G$. In 1984,  Goldberg conjectured that for any multigraph $G$, if $\chi'(G) \ge \Delta(G) +3$ then $\chi''(G) = \chi'(G)$. We show that Goldberg's conjecture is asymptotically true. More specifically,  we prove that for a multigraph $G$ with maximum degree $\Delta$ sufficiently large, $\chi''(G) = \chi'(G)$ provided $\chi'(G) \ge \Delta + 10\Delta^{35/36}$.  When $\chi'(G) \ge \Delta(G) +2$, the chromatic index $\chi'(G)$ is completely determined by the fractional chromatic index. Consequently,   the total chromatic number $\chi''(G)$ can be computed in polynomial-time in this case.

The burning number $b(G)$ of a graph $G$ is the smallest integer $k$ such that $G$ can be covered by $k$ balls of radii respectively $0,\dots,k-1$, and was introduced independently by Brandenburg and Scott at Intel as a transmission problem on processors \cite{alon} and Bonato, Janssen and Roshanbin as a model for the spread of information in social networks.

The Burning Number Conjecture \cite{bonato} claims that $b(G)\leq \left\lceil\sqrt{n}\right\rceil$, where $n$ is the number of vertices of $G$. This bound tight for paths. The previous best bound for this problem, by Bastide et al. \cite{bastide}, was $b(G)\leq \sqrt{\frac{4n}{3}}+1$.

We prove that the Burning Number Conjecture holds asymptotically, that is $b(G)\leq (1+o(1))\sqrt{n}$.

Following a brief introduction to graph burning, this talk will focus on the general ideas behind the proof.

Given a graph G with edges labeled by a group, a construction of Zaslavsky gives a rank-1 lift of the graphic matroid M(G) that respects the group-labeling. For which finite groups can we construct a rank-t lift of M(G) with t > 1 that respects the group-labeling? We show that this is possible if and only if the group is the additive subgroup of a non-prime finite field. We assume no knowledge of matroid theory.

An order-n Latin square is an $n \times n$ matrix with entries from a set of $n$ symbols, such that each row and each column contains each symbol exactly once.  Suppose that $L_{i,j} \subseteq [n]$ is a random subset of $[n]$ where each $k \in [n]$ is included in $L_{i,j}$ independently with probability $p$ for each $i,j\in[n]$.  How likely does there exist an order-$n$ Latin square where the entry in the $i$th row and $j$th column lies in $L_{i,j}$?  This question was initially raised by Johansson in 2006, and later Casselgren and H{\"a}ggkvist and independently Luria and Simkin conjectured that $\log n / n$ is the threshold for this property.  In joint work with Dong-yeap Kang, Daniela K\"{u}hn, Abhishek Methuku, and Deryk Osthus, we proved that for some absolute constant $C$, if $p > C \log^2 n / n$, then asymptotically almost surely there exists such a Latin square.  We also prove analogous results for Steiner triple systems and $1$-factorizations of complete graphs.