Let \(G\) be a finite and simple graph with vertex set \(V(G)\) and edge set \(E(G)\). The arboricity \(\Upsilon(G)\) of \(G\) is the minimum number \(k\) such that the edges of \(G\) can be partitioned into \(k\) forests. The diameter-\(d\) arboricity \(\Upsilon_d(G)\) of \(G\) is the minimum number \(k\) such that the edge of \(G\) can be partitioned into \(k\) forests each of whose components have diameter at most \(d\), and the bounded diameter arboricity of a class of graphs is the minimum number \(k\) such that, for some positive integer \(d\), every graph in the class has diameter-\(d\) arboricity at most \(k\).
The authors conjecture that the class of graphs with arboricity at most \(k\) has bounded diameter arboricity at most \(k+1\); they prove the conjecture for \(k \in \{2,3\}\) by showing that the union of a forest and a star forest can be partitioned into two forests in which every tree has diameter at most 18.
The authors also conjecture that the planar graphs of girth at least 5 have bounded diameter arboricity at most 2; they prove this conjecture for girth at least 6 using the result of S.-J. Kim et al. [ibid. 74, No. 3–4, 369–391 (2013; Zbl 1276.05092)]. They also show that the planar graphs of girth at least 3 (4, respectively) have bounded diameter arboricity at most 4 (3, respectively).
For a planar graph \(G\) and for a set \(A \subseteq V(G)\), let \(\sigma_G(A)=\{ab \in E(G): a \in A \text{ and } b \not\in A\}\). For a real number \(\varepsilon\) with \(0< \varepsilon <1\), a spanning subgraph \(H\) of \(G\) is called \(\varepsilon\)-thin if \(|\sigma_H(A)| \le \varepsilon |\sigma_G(A)|\) for every \(A \subseteq V(G)\). The authors show that every 6-edge-connected planar (multi)graph contains two edge-disjoint \(\frac{18}{19}\)-thin spanning trees.
The authors also show that, for some \(d \ge 4\), \(\Upsilon(G)=k\) implies \(\Upsilon_d(G) \le \frac{4}{3}k\) for large enough \(k\), and thus answering the question raised in [T. Mütze and U. Peter, Discrete Math. 313, No. 22, 2626–2637 (2013; Zbl 1281.05088)]. This result leads to an improved lower bound on the density of globally sparse Ramsey graphs.