Linear arboricity

In graph theory, a branch of mathematics, the linear arboricity of an undirected graph is the smallest number of linear forests its edges can be partitioned into. Here, a linear forest is an acyclic graph with maximum degree two; that is, it is a disjoint union of path graphs. Linear arboricity is a variant of arboricity, the minimum number of forests into which the edges can be partitioned.

The linear arboricity of any graph of maximum degree ${\displaystyle \Delta }$ is known to be at least ${\displaystyle \lceil \Delta /2\rceil }$ and is conjectured to be at most ${\displaystyle \lceil (\Delta +1)/2\rceil }$. This conjecture would determine the linear arboricity exactly for graphs of odd degree, as in that case both expressions are equal. For graphs of even degree it would imply that the linear arboricity must be one of only two possible values, but determining the exact value among these two choices is NP-complete.