Georgia Institute of Technology College of Sciences

A cubic graph is one where every vertex has degree three. A linear forest is a disjoint union of paths. It is known that the edge set of every cubic graph can be partitioned into two linear forests where each path is short (of constant size). A conjecture of Wormald asks for such a partition where the two forests are isomorphic (we no longer insist on having short paths, although that is also an open question). Note that this also can be phrased as an edge-colouring question. Is it possible to colour the edge set of a cubic graph by red and blue such that the two monochromatic components induce isomorphic linear forests? Recently we proved this for all connected graphs on a sufficiently large number of vertices. I will talk about the result and give some idea of the proof method. This is joint work with Gal Kronenberg, Shoham Letzter and Alexey Pokrovskiy.

We define a graph process based on a discrete branching process with deletions and mergers, which is inspired by the 4-cycle structure of the hypercube $\mathcal{Q}_d$ for large $d$. We prove survival and extinction under certain conditions on $p$ and $q$ that heuristically match the known expansions of the critical probabilities for bond percolation on the hypercube. Joint work with Laura Eslava and Sarah Penington. Based on https://arxiv.org/abs/2104.04407.

We discuss a few new results concerning the descriptive combinatorics of bounded degree hyperfinite Borel graphs. In particular, we show that the Baire measurable edge chromatic number of $G$ is at most $\lceil\frac{3}{2}\Delta(G)\rceil+6$ when G is a multigraph, and for bipartite graphs we improve this bound to $\Delta(G)+1$ and show that degree regular one-ended bipartite graphs have Borel perfect matchings generically. Similar results hold in the measure setting assuming some hyperfiniteness conditions. This talk is based on joint work with Kun and Sabok, Weilacher, and upcoming work with Poulin and Zomback.

In his works Carlitz defined and investigated a few generalizations of the Eulerian numbers and polynomials. For most of those generalizations he provided also a combinatorial interpretation. The classical Eulerian numbers and some of their generalizations are connected to combinatorial statistics on permutations. Carlitz intended to provide a combinatorial interpretation also to his degenerate Eulerian numbers. However since their introduction in 1979 these numbers had a pure analytic character. In this talk we consider a combinatorial model that generalizes the standard definition of permutations and show its relation to the degenerate Eulerian numbers.