YouTube recording: https://www.youtube.com/watch?v=3KYEAEiHkb0
Time: Thursday, Apr 27, 3pm ET
Speaker: Sebastian Mies, Johannes Gutenberg University Mainz
Title: The Strong Nine Dragon Tree Conjecture for $d \le k+1$
Abstract: The arboricity $\Gamma(G)$ of an undirected graph $G = (V,E)$ is the minimal number such that $E$ can be partitioned into $\Gamma(G)$ forests. Nash-Williams’ formula states that $\Gamma(G) = \lceil \gamma(G) \rceil$, where $\gamma(G)$ is the maximum of $|E_H|/(|V_H|-1)$ over all subgraphs $(V_H, E_H)$ of $G$ with $|V_H| \ge 2$. The Strong Nine Dragon Tree Conjecture states that if $\gamma(G) \le k + d / (d+k+1)$ for natural numbers k, d, then there is a partition of the edge set of G into k+1 forests such that one forest has at most d edges in each connected component. We settle the conjecture for $d \le k + 1$. For $d \le 2(k+1)$, we cannot prove the conjecture, however we show that there exists a partition in which the connected components in one forest have at most $d + \lceil kd/(k+1) \rceil – k$ edges. As an application of this theorem, we show that every 5-edge-connected planar graph G has a 5/6-thin spanning tree, a spanning tree whose edges fill up at most 5/6 of every cut. This theorem is best possible, in the sense that we cannot replace 5-edge-connected with 4-edge-connected, even if we replace 5/6 with any positive real number less than 1. This strengthens a result of Merker and Postle which showed 6-edge-connected planar graphs have a 18/19-thin spanning tree. This is joint work with Benjamin Moore.