Unified bijections for planar hypermaps with general cycle-length constraints

We present a general bijective approach to planar hypermaps with two main results. First we obtain unified bijections for all classes of maps or hypermaps defined by face-degree constraints and girth constraints. To any such class we associate bijectively a class of plane trees characterized by local constraints. This unifies and greatly generalizes several bijections for maps and hypermaps. Second, we present yet another level of generalization of the bijective approach by considering classes of maps with non-uniform girth constraints. More precisely, we consider "well-charged maps", which are maps with an assignment of "charges" (real numbers) on vertices and faces, with the constraints that the length of any cycle of the map is at least equal to the sum of the charges of the vertices and faces enclosed by the cycle. We obtain a bijection between charged hypermaps and a class of plane trees characterized by local constraints

The Strong Nine Dragon Tree Conjecture is true for d≤k+1d \leq k+1

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 $k = \lceil \gamma(G) \rceil$, where $\gamma(G)$ is the maximum of ${|E_H|}/(|V_H| -1)$ over all subgraphs $(V_H, E_H)$ of $G$ with $|V_H| \geq 2$. The Strong Nine Dragon Tree Conjecture states that if $\gamma(G) \leq k + \frac{d}{d+k+1}$ for $k, d \in \mathbb N_0$, 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 $d \leq k + 1$. For $d \leq 2(k+1)$, we cannot prove the conjecture, however we show that there exists a partition in which the connected components in one forest have at most $d + \lceil k \cdot \frac{d}{k+1} \rceil - k$ edges. As an application of this theorem, we show that every $5$-edge-connected planar graph $G$ has a $\frac{5}{6}$-thin spanning tree. This theorem is best possible, in the sense that we cannot replace $5$-edge-connected with $4$-edge-connected, even if we replace $\frac{5}{6}$ 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 $\frac{18}{19}$-thin spanning tree.Comment: 24 pages, paper updated in accordance to referee comment