Linear forest

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]

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]

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 Δ {\displaystyle \Delta } , the linear arboricity is always at least ⌈ Δ / 2 ⌉ {\displaystyle \lceil \Delta /2\rceil } , and it is conjectured that it is always at most ⌊ ( Δ + 1 ) / 2 ⌋ {\displaystyle \lfloor (\Delta +1)/2\rfloor } .^[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 Δ 3 / 2 {\displaystyle \Delta ^{3/2}} , and there exist graphs for which it is at least proportional to this quantity.^[12]