Bounded Diameter Arboricity

TL;DR

This paper introduces bounded diameter arboricity, a measure of how a graph's edges can be partitioned into forests with limited component diameters, and proves related conjectures for specific cases, with applications to planar graphs and spanning trees.

Contribution

It defines bounded diameter arboricity, proves the conjecture for k=2,3, and characterizes this property for certain planar graphs, advancing understanding of graph decompositions.

Findings

Proved the conjecture for k=2,3.

Characterized bounded diameter arboricity for planar graphs with certain girths.

Showed existence of two edge-disjoint thin spanning trees in 6-edge-connected planar graphs.

Abstract

We introduce the notion of \emph{bounded diameter arboricity}. Specifically, the \emph{diameter-$d$ arboricity} of a graph is the minimum number $k$ such that the edges of the graph can be partitioned into $k$ forests each of whose components has diameter at most $d$. A class of graphs has bounded diameter arboricity $k$ if there exists a natural number $d$ such that every graph in the class has diameter-$d$ arboricity at most $k$. We conjecture that the class of graphs with arboricity at most $k$ has bounded diameter arboricity at most $k+1$. We prove this conjecture for $k\in \{2,3\}$ by proving the stronger assertion that the union of a forest and a star forest can be partitioned into two forests of diameter at most 18. We use these results to characterize the bounded diameter arboricity for planar graphs of girth at least $g$ for all $g\ne 5$. As an application we show that every 6-edge-connected planar (multi)graph contains two edge-disjoint $\frac{18}{19}$-thin spanning trees.