Linear forest
Graph formed from disjoint paths
From Wikipedia, the free encyclopedia
In graph theory, a branch of mathematics, a linear forest is a kind of forest where each component is a path graph,[1] or a disjoint union of nontrivial paths.[2] Equivalently, it is an acyclic and claw-free graph.[3] An acyclic graph where every vertex has degree 0, 1, or 2 is a linear forest.[4][5] An undirected graph has Colin de Verdière graph invariant at most 1 if and only if it is a (node-)disjoint union of paths, i.e. it is linear.[6][7] Any linear forest is a subgraph of the path graph with the same number of vertices.[8]
Extensions to the notation
According to Habib and Peroche, a k-linear forest consists of paths of k or fewer nodes each.[9]
According to Burr and Roberts, an (n, j)-linear forest has n vertices and j of its component paths have an odd number of vertices.[2]
According to Faudree et al., a (k, t)-linear or (k, t, s)-linear forest has k edges, and t components of which s are single vertices; s is omitted if its value is not critical.[10]
Derived concepts
The linear arboricity of a graph is the minimum number of linear forests into which the graph can be partitioned. For a graph of maximum degree , the linear arboricity is always at least , and it is conjectured that it is always at most .[11]
A linear coloring of a graph is a proper graph coloring in which the induced subgraph formed by each two colors is a linear forest. The linear chromatic number of a graph is the smallest number of colors used by any linear coloring. The linear chromatic number is at most proportional to , and there exist graphs for which it is at least proportional to this quantity.[12]
