Related papers: New bounds for linear arboricity and related probl…
The linear arboricity of a graph $G$, denoted by $\text{la}(G)$, is the minimum number of edge-disjoint linear forests (i.e. forests in which every connected component is a path) in $G$ whose union covers all the edges of $G$. A famous…
A linear forest is a union of vertex-disjoint paths, and the linear arboricity of a graph $G$, denoted by $\operatorname{la}(G)$, is the minimum number of linear forests needed to partition the edge set of $G$. Clearly,…
In 1980, Akiyama, Exoo and Harary posited the Linear Arboricity Conjecture which states that any graph $G$ of maximum degree $\Delta$ can be decomposed into at most $\left\lceil \frac{\Delta}{2}\right\rceil$ linear forests. (A forest is…
The Linear Arboricity Conjecture asserts that the linear arboricity of a graph with maximum degree $\Delta$ is $\lceil (\Delta+1)/2 \rceil$. For a $2k$-regular graph $G$, this implies $la(G) = k+1$. In this note, we utilize a network flow…
The linear arboricity la(G) of a graph G is the minimum number of linear forests that partition the edges of G. In 1984, Akiyama et al. stated the Linear Arboricity Conjecture (LAC), that the linear arboricity of any simple graph of maximum…
A (directed) linear forest is a (di)graph whose components are (directed) paths. The linear arboricity $la(F)$ of a (di)graph $F$ is the minimum number of (directed) linear forests required to decompose its edges. Akiyama, Exoo, and Harary…
A linear forest is an acyclic graph whose each connected component is a path; or in other words, it is an acyclic graph whose maximum degree is at most 2. A linear coloring of a graph $G$ is an edge coloring of $G$ such that the edges in…
A linear forest is a forest in which every connected component is a path. The linear arboricity of a graph $G$ is the minimum number of linear forests of $G$ covering all edges. In 1980, Akiyama, Exoo and Harary proposed a conjecture, known…
In 1980, Akiyama, Exoo, and Harary conjectured that any graph $G$ can be decomposed into at most $\lceil(\Delta(G)+1)/2\rceil$ linear forests. We confirm the conjecture for robust expanders of linear minimum degree. As a consequence, the…
Wu, Zhang and Li [4] conjectured that the set of vertices of any simple graph $G$ can be equitably partitioned into $\lceil(\Delta(G)+1)/2\rceil$ subsets so that each of them induces a forest of $G$. In this note, we prove this conjecture…
A path partition (also referred to as a linear forest) of a graph $G$ is a set of vertex-disjoint paths which together contain all the vertices of $G$. An isolated vertex is considered to be a path in this case. The path partition…
We give an algorithm for finding the arboricity of a weighted, undirected graph, defined as the minimum number of spanning forests that cover all edges of the graph, in $\sqrt{n} m^{1+o(1)}$ time. This improves on the previous best bound of…
We show that the edges of any planar graph of maximum degree at most $9$ can be partitioned into $4$ linear forests and a matching. Combined with known results, this implies that the edges of any planar graph $G$ of odd maximum degree…
The \emph{linear vertex arboricity} of a graph is the smallest number of sets into which the vertices of a graph can be partitioned so that each of these sets induces a linear forest. Chaplick et al. [JoCG 2020] showed that, somewhat…
A \emph{linear $k$-forest} is a forest whose components are paths of length at most $k$. The \emph{linear $k$-arboricity} of a graph $G$, denoted by ${\rm la}_k(G)$, is the least number of linear $k$-forests needed to decompose $G$.…
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…
Let $G$ be a graph on $n$ vertices. A linear forest is a graph consisting of vertex-disjoint paths and isolated vertices. A maximum linear forest of $G$ is a subgraph of $G$ with maximum number of edges, which is a linear forest. We denote…
A \emph{tree-partition} of a graph $G$ is a proper partition of its vertex set into `bags', such that identifying the vertices in each bag produces a forest. The \emph{tree-partition-width} of $G$ is the minimum number of vertices in a bag…
For some $k \in \mathbb{Z}_{\geq 0}\cup \infty$, we call a linear forest $k$-bounded if each of its components has at most $k$ edges. We will say a $(k,\ell)$-bounded linear forest decomposition of a graph $G$ is a partition of $E(G)$ into…
We prove that for any positive integer $k$, the edges of any graph whose fractional arboricity is at most $k + 1/(3k+2)$ can be decomposed into $k$ forests and a matching.