In graph theory, a branch of mathematics, a linear forest is a kind of forest where each component is a path graph,: 200 or a disjoint union of nontrivial paths.: 246 Equivalently, it is an acyclic and claw-free graph.: 130, 131 An acyclic graph where every vertex has degree 0, 1, or 2 is a linear forest.: 310 : 107 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.: 13, 19–21 : 29, 35, 67 (3, 6, 29) Any linear forest is a subgraph of the path graph with the same number of vertices.: 55
Extensions to the notation
According to Habib and Peroche, a k-linear forest consists of paths of k or fewer nodes each.9: 219
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.10: 245, 246
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.11: 1178, 1179
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 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 } .12
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.13
References
Harary, Frank (September 1970). "Covering and Packing in Graphs, I". Annals of the New York Academy of Sciences. 175 (1): 198–205. Bibcode:1970NYASA.175..198H. doi:10.1111/j.1749-6632.1970.tb56470.x. /wiki/Frank_Harary ↩
Burr, Stefan A.; Roberts, John A. (May 1974). "On Ramsey numbers for linear forests". Discrete Mathematics. 8 (3). North-Holland Publishing Company: 245–250. doi:10.1016/0012-365x(74)90136-8. MR 0335325. /wiki/Stefan_Burr ↩
Brandstädt, Andreas; Giakoumakis, Vassilis; Milanič, Martin (2018-12-11). "Weighted efficient domination for some classes of H-free and of (H1,H2)-free graphs". Discrete Applied Mathematics. 250. Elsevier B.V.: 130–144. doi:10.1016/j.dam.2018.05.012. MR 3868662. EBSCOhost 45704539, 132688071. /wiki/Andreas_Brandst%C3%A4dt ↩
Enomoto, Hikoe; Péroche, Bernard (Summer 1984). "The Linear Arboricity of Some Regular Graphs". Journal of Graph Theory. 8 (2): 309–324. doi:10.1002/jgt.3190080211. /wiki/Doi_(identifier) ↩
Jain, Sparsh; Pallathumadam, Sreejith K.; Rajendraprasad, Deepak (February 10–12, 2022). "B0-VPG Representation of AT-free Outerplanar Graphs". Written at Puducherry, India. In Balachandran, Niranjan; Inkulu, R. (eds.). Algorithms and Discrete Applied Mathematics: 8th International Conference, CALDAM 2022. Lecture Notes in Computer Science. Vol. 13179. Cham, Switzerland: Springer Nature. pp. 103–114. doi:10.1007/978-3-030-95018-7_9. ISBN 978-3-030-95017-0. 978-3-030-95017-0 ↩
de Verdière, Yves Colin (October 1990). "Sur un Nouvel Invariant des Graphes et un Critère de Planarité". Journal of Combinatorial Theory, Series B (in French). 50 (1). Academic Press, Inc.: 11–21. doi:10.1016/0095-8956(90)90093-F. /wiki/Yves_Colin_de_Verdi%C3%A8re ↩
van der Holst, Hein; Lovász, László; Schrijver, Alexander (1999) [Preliminary version, March 1997]. "The Colin de Verdière graph parameter". In Lovász, László; Gyárfás, András; Katona, Gyula; Recski, András; Székely, László (eds.). Graph Theory and Combinatorial Biology. Bolyai Society Mathematical Studies. Vol. 7. Budapest, Hungary: János Bolyai Mathematical Society (Hungarian: Bolyai János Matematikai Társulat). pp. 29–85 [1–43]. ISBN 963-8022-90-6. MR 1673503. Archived from the original on 2007-02-06. Associated with the "International Colloquium on Combinatorics and Graph Theory" in Balatonlelle on July 1996 (see p. 5 and http://wwwold.sztaki.hu/library/publtop/gyarf.jhtml). {{cite book}}: External link in |postscript= (help)CS1 maint: postscript (link) 963-8022-90-6 ↩
Clark, Curtis (1984). An Approach to Graph Achievement Games: Ultimately Economical Graphs (PhD thesis). Ann Arbor, Michigan: University of Michigan. ISBN 979-8-204-34535-5. ProQuest 303324911 (UMI 8502782) – via ProQuest Dissertations & Theses Global. 979-8-204-34535-5 ↩
Habib, M.; Peroche, B. (1982). "Some problems about linear arboricity". Discrete Mathematics. 41 (2). North-Holland Publishing Company: 219–220. doi:10.1016/0012-365x(82)90209-6. https://doi.org/10.1016%2F0012-365x%2882%2990209-6 ↩
Burr, Stefan A.; Roberts, John A. (May 1974). "On Ramsey numbers for linear forests". Discrete Mathematics. 8 (3). North-Holland Publishing Company: 245–250. doi:10.1016/0012-365x(74)90136-8. MR 0335325. /wiki/Stefan_Burr ↩
Faudree, Ralph J.; Gould, Ronald J.; Jacobson, Michael S. (28 March 2009). "Pancyclic graphs and linear forests". Discrete Mathematics. 309 (5). Elsevier B.V.: 1178–1189. doi:10.1016/j.disc.2007.12.094. /wiki/Ralph_Faudree ↩
Alon, N. (1988), "The linear arboricity of graphs", Israel Journal of Mathematics, 62 (3): 311–325, CiteSeerX 10.1.1.163.1965, doi:10.1007/BF02783300, MR 0955135. /wiki/Noga_Alon ↩
Yuster, Raphael (1998), "Linear coloring of graphs", Discrete Mathematics, 185 (1–3): 293–297, doi:10.1016/S0012-365X(97)00209-4, MR 1614290. /wiki/Discrete_Mathematics_(journal) ↩