degenerate graphslinear arboricitylinear forest partitionA linear forest is a union of vertex‐disjoint paths, and the linear arboricity of a graph G $G$ , denoted by la ( G ) $\\,ext{la}(G)$ , is the minimum n
Given a graph G, we define its linear arboricity, denoted by la(G), to be the minimum number of edge-disjoint linear forests in G whose union is E(G). This notion was introduced by Harary [16] in 1970 as one of the covering invariants of graphs, and has been studied quite ...
A linear forest is a union of vertex‐disjoint paths, and the linear arboricity of a graph G $G$, denoted by la(G) $\\,ext{la}(G)$, is the minimum number of linear forests into which the edge set of G $G$ can be partitioned. Clearly, la(G)≥Δ(G)∕2 $\\,ext{la}(G)...
We prove the conjecture for 3-degenerate graphs. This establishes the conjecture for graphs of treewidth at most 3 and provides an alternative proof for the conjecture for triangle-free planar graphs. Our proof also yields an $O(n)$-time algorithm that partitions the edge set of any 3-...
In particular, we show that the Linear Arboricity Conjecture holds for k $k$‐degenerate loopless multigraphs when the maximum degree is at least 4k2 $4k-2$, improving a recent bound by Chen, Hao, and Yu for simple graphs. Finally, we demonstrate that the (g,h) $(g,h)$‐oriented ...
2-degenerateThe linear 2-arboricity la2(G) of a graph G is the least integer k such that G can be partitioned into k edge-disjoint forests, whose components are paths of length at most 2. In this paper, we study the linear 2-arboricity of sparse graphs, and prove the following ...
A linear forest is a disjoint union of path graphs. The linear arboricity of a graphG $G$, denoted by la(G) $ext{la}(G)$, is the least number of linear forests into which the graph can be partitioned. Clearly, la(G)≥Δ(G)∕2 $ext{la}(G)\\ge \\lceil {m{\\Delta }}(...
k-degenerate graphDense random graphsA graph is a linear forest if each of its components is a path. Given a graph G with maximum degree Delta(G), motivated by the famous linear arboricity conjecture and Lovasz's classic result on partitioning the edge set of a graph into paths, we call...