Abstract: We consider the question of determining the minimum number of pentagons in a linear triple system with many edges and show some connections to number theory, graph theory, theoretical computer science,  and geometry. This is joint work with Jozsef Solymosi.

Abstract: A Berge cycle of length $k$ in a hypergraph is a set of $k$ distinct vertices $v_1, \ldots, v_k$ and $k$ distinct edges $e_1, \ldots, e_k$ such that $v_i, v_{i+1} \in e_i$ (with indices taken modulo $k$). In this talk, we discuss various extremal results for Berge cycles in hypergraphs, strengthening classical results for cycles in graphs. 

Abstract: TBA

Abstract: The rectilinear crossing number of a graph is the minimum number of edge crossings in a straight-line drawing of the graph in the plane. Despite its elementary definition, even the case of complete graphs remains poorly understood, and progress over the past several decades has relied on a blend of geometric intuition, combinatorial reasoning, and increasingly sophisticated computational methods. In this talk, I will present a progress report on the use of artificial intelligence as a new exploratory tool for this classical problem. After briefly surveying the history of the rectilinear crossing number and its known constructions, I will introduce AlphaEvolve, which has recently been applied to the study of mathematical conjectures. I will describe the framework at a high level and explain how it was adapted to the rectilinear crossing number. I will then discuss results from this ongoing work. AlphaEvolve was able to improve upon the best known constructions for 55 complete graphs. I conclude by reflecting on the potential role of AI-assisted exploration in mathematics. This is joint work with Weonyoung Joo, Jeong Han Kim, and Bernard Lidicky.

Abstract: Dirac’s classical theorem asserts that, for $n\ge 3$, any $n$-vertex graph with minimum degree at least $n/2$ is Hamiltonian.  Furthermore, if we additionally assume that such graphs are regular, then, by the breakthrough work of Csaba, Kühn, Lo, Osthus and Treglown, they admit a decomposition into Hamilton cycles and at most one perfect matching, solving the well-known Nash‑Williams conjecture.

In the pseudorandom setting, it has long been conjectured that similar results hold in much sparser graphs.

We prove two theorems for graphs that exclude excessively dense subgraphs, which yield asymptotically optimal resilience and Hamilton‑decomposition results in sparse pseudorandom graphs. In particular, our results imply that for every fixed $\gamma>0$, there exists a constant $C>0$ such that if $G$ is a spanning subgraph of an $(n,d,\lambda)$-graph satisfying $\delta(G)\ge\bigl(\tfrac{1}{2}+\gamma\bigr)d$ and $d/\lambda\ge C,$ then $G$ must contain a Hamilton cycle.

Secondly, we show that for every $\varepsilon>0$, there is $C>0$ so that any $(n,d,\lambda)$-graph with $d/\lambda\ge C$ contains at least $\bigl(\tfrac{1}{2}-\varepsilon\bigr)d$ edge‑disjoint Hamilton cycles, and, finally, we prove that the entire edge set of $G$ can be covered by no more than $\bigl(\tfrac12+\varepsilon\bigr)d$ such cycles. All bounds are asymptotically optimal and  significantly improve earlier results on Hamiltonian resilience, packing, and covering in sparse pseudorandom graphs. This is joint work with Nemanja Draganic, Hyunwoo Lee, David Munha Correia, Matias Pavez-Signe and Benny Sudakov.

Abstract: TBA

Abstract: A graph $G$ is $(a,b)$-sparse if every nonempty subgraph $H$ satisfies $e(H) \leq a v(H) + b$.  We will discuss conditions under which an $(a,b)$-sparse graph can be partitioned $E(G) = E(G_1) \cup E(G_2)$ such that for $i \in \{1,2\}$ we have that $G_i$ is $(a_i, b_i)$-sparse.  Then we will apply this result to the Kohayakawa--Kreuter Conjecture for Families, which asks when a random graph is Ramsey.

Abstract: An arc-weighted orientation of a graph $G$ is a pair $(D,w)$, where $D$ is an orientation of $G$ and $w$ is a weight assignment that assigns to each arc a positive integer as its weight. We say $(D,w)$ is acyclic if for every non-empty sub-digraph $D'$ of $D$, $(D',w)$ has a dominating arc, i.e., an arc $e=(u,v)$ with $w(e) > d_{(D',w)}^+(v)$. A graph $G$ is strict AW-$f$-degenerate if $G$ has an arc-weighted acyclic orientation $(D,w)$ with $d_{(D,w)}^+(v) < f(v)$ for each vertex $v$. We show that the strict AW-degeneracy of $G$ is an upper bound for many colouring parameters, including the DP-paint number and AT number. A graph $G$ is degree-truncated $k$-choosable if $G$ is $f$-choosable with $f(v) = \min\{k, d_G(v)\}$.  We show that there are 3-connected non-complete planar graphs  that are not  degree-truncated $9$-choosable (that answers in negative a question of Richter) and every 3-connected non-complete planar graph is strict AW-weighted degree-truncated $15$-degenerate and hence degree-truncated $15$-choosable.

Abstract: For a non-decreasing sequence $S = (s_1, s_2, \ldots, s_k)$ of positive integers, a packing $S$-coloring of a graph $G$ is a partition of $V(G)$ into $V_1, V_2, \ldots, V_k$ such that each $V_i$ has pairwise distance at least $s_i+1$. The packing chromatic number (PCN) of a graph $G$ is the minimum $k$ such that $G$ has a packing $(1,2, \ldots, k)$-coloring. The $1$-subdivision of $G$ is obtained by replacing each edge of $G$ with a path of two edges. In 2016, Gastineau and Togni asked an open question whether the $1$-subdivision of every subcubic graph has PCN at most $5$, and later Bre\v sar, Klav\v zar, Rall, and Wash conjectured it is true. Balogh, Kostochka, and Liu proved the first upper bound of $8$, and it was later improved to $6$ by Liu, Zhang, and Zhang. 

We prove that every connected subcubic graph except the Petersen graph is packing $(1,1,2,2)$-colorable. Our result implies a solution to the conjecture of Bre\v sar, Klav\v zar, Rall, and Wash, and answers the question of Gastineau and Togni in the affirmative. Furthermore, our result answers an open question of Kostochka and Liu and solves a conjecture of Liu, Zhang, and Zhang. This is a joint work with Xinmin Hou and Xiangyang Wang.

Abstract: Let $G$ be a graph and $L$ be a list-assignment for $G$. The \emph{reconfiguration graph} $\mathcal{C}_L(G)$  has as its vertices the proper $L$-colorings of $G$, with edges joining $L$-colorings that differ on a single vertex. Cambie, Cames van Batenburg, and the first author conjectured that if $d:=\mad(G)$ and $k\ge \lceil d+2\rceil$, then for each $k$-assignment $L$ we have $\diam~\C_L(G) = O_d(|G|)$.  This conjecture is easy to verify when $d\le 2$.  We prove the conjecture when $d\in (2,3)\cup (3,3.75)$, strengthening results of the first author, who handled the case $d\in (3,3.4)$.  Specifically, for each $\epsilon >0$, we prove that if $\mad(G)<3-\epsilon$ and $L$ is a 5-assignment for $G$, then $\diam~\C_L(G)=O_{\epsilon}(|G|)$.  Similarly, if $\mad(G)<3.75-\epsilon$ and $L$ is a 6-assignment for $G$, then $\diam~\C_L(G)=O_{\epsilon}(|G|)$.  The key new tool in our proofs is a \emph{Generalized Extension Lemma}, which allows us to consider reducible configuration of arbitrary size.  To best leverage this power, our proofs rely heavily on discharging arguments that are fundamentally \emph{global}, rather than local.  This is a joint work with Dan Cranston and Xuzhong Wang.

Abstract: The {\it $k$-deck} of an $n$-vertex graph $G$ is its multiset of subgraphs induced by $k$ vertices.  The famous Reconstruction Conjecture of Kelly and Ulam states that the $(n-1)$-deck determines $G$ when $n\ge3$.  Always the $k$-deck determines the $(k-1)$-deck, so an enhanced version of the reconstruction problem is to seek the largest $\ell$ such that the $(n-\ell)$-deck determines $G$.

An $n$-vertex graph is {\it $\ell$-reconstructible} if it is determined by its $(n-\ell)$-deck.  Manvel conjectured that for each $\ell$ there is a threshold $M_\ell$ such that every graph with at least $M_\ell$ vertices is $\ell$-reconstructible; the Reconstruction Conjecture is $M_1=3$.

The first part of this talk describes known results on the general problem.

Of particular interest is the threshold on $n$ to guarantee $\ell$-reconstructibility of $n$-vertex trees.  It is known that $n\ge6\ell+11$ suffices, and it is conjectured that $n\ge2\ell+1$ suffices, with small exceptions.  The second part of the talk presents several ideas from a recent proof that in general for caterpillars $n\ge2\ell+1$ suffices and is best possible (caterpillars are the trees in which deleting the leaves produces a path).  The result on caterpillars is joint work with Alexandr V. Kostochka, Zishen Qu, and Maddy Ritter.

Abstract: TBA