Linear forest

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Graph formed from disjoint paths

title: "Linear forest" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["trees-(graph-theory)", "graph-families"] description: "Graph formed from disjoint paths" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Linear_forest" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Graph formed from disjoint paths ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/7/77/Linear_forest.svg" caption="cycles"] ::

In graph theory, a branch of mathematics, a linear forest is a kind of forest where each component is a path graph, or a disjoint union of nontrivial paths. Equivalently, it is an acyclic and claw-free graph. An acyclic graph where every vertex has degree 0, 1, or 2 is a linear forest. 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. Any linear forest is a subgraph of the path graph with the same number of vertices.

Extensions to the notation

According to Habib and Peroche, a k-linear forest consists of paths of k or fewer nodes each.

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.

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.

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 \Delta, the linear arboricity is always at least \lceil\Delta/2\rceil, and it is conjectured that it is always at most \lfloor(\Delta+1)/2\rfloor.{{citation | last = Alon | first = N. | authorlink = Noga Alon | doi = 10.1007/BF02783300 | doi-access = free | issue = 3 | journal = Israel Journal of Mathematics | mr = 955135 | pages = 311–325 | title = The linear arboricity of graphs | volume = 62 | year = 1988| citeseerx = 10.1.1.163.1965 }}.

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 \Delta^{3/2}, and there exist graphs for which it is at least proportional to this quantity.{{citation | last = Yuster | first = Raphael | author-link = Raphael Yuster | doi = 10.1016/S0012-365X(97)00209-4 | issue = 1–3 | journal = Discrete Mathematics | mr = 1614290 | pages = 293–297 | title = Linear coloring of graphs | volume = 185 | year = 1998| doi-access = free

References

References

  1. Harary, Frank. (September 1970). "Covering and Packing in Graphs, I". Annals of the New York Academy of Sciences.
  2. (May 1974). "On Ramsey numbers for linear forests". North-Holland Publishing Company.
  3. (2018). "Weighted efficient domination for some classes of H-free and of (H1,H2)-free graphs". [[Elsevier]] B.V..
  4. (1984). "The Linear Arboricity of Some Regular Graphs". Journal of Graph Theory.
  5. (February 10–12, 2022). "Algorithms and Discrete Applied Mathematics: 8th International Conference, CALDAM 2022". Springer Nature.
  6. de Verdière, Yves Colin. (October 1990). "Sur un Nouvel Invariant des Graphes et un Critère de Planarité". Academic Press, Inc..
  7. (1999). "Graph Theory and Combinatorial Biology". [[János Bolyai Mathematical Society]].
  8. Clark, Curtis. (1984). "An Approach to Graph Achievement Games: Ultimately Economical Graphs". University of Michigan.
  9. (1982). "Some problems about linear arboricity". North-Holland Publishing Company.
  10. (28 March 2009). "Pancyclic graphs and linear forests". Elsevier B.V..

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