Thickness (graph theory)

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In graph theory, the thickness of a graph G is the minimum number of planar graphs into which the edges of G can be partitioned. That is, if there exists a collection of k planar graphs, all having the same set of vertices, such that the union of these planar graphs is G, then the thickness of G is at most k.[1]Cite error: Closing </ref> missing for <ref> tag and on a related 1962 conjecture of Frank Harary: For any graph on 9 points, either itself or its complementary graph is non-planar. The problem is equivalent to determining whether the complete graph K9 is biplanar (it is not, and the conjecture is true).[2] A comprehensive[3] survey on the state of the arts of the topic as of 1998 was written by Petra Mutzel, Thomas Odenthal and Mark Scharbrodt.[4]

Specific graphs

The thickness of the complete graph on n vertices, Kn, is

⌊n+76⌋,

except when n = 9, 10 for which the thickness is three.[5][6]

With some exceptions, the thickness of a complete bipartite graph Ka,b is generally:[7][8]

⌈ab2(a+b−2)⌉.

Properties

Every forest is planar, and every planar graph can be partitioned into at most three forests. Therefore, the thickness of any graph G is at most equal to the arboricity of the same graph (the minimum number of forests into which it can be partitioned) and at least equal to the arboricity divided by three.[4][9]

The graphs of maximum degree d have thickness at most ⌈d/2⌉.[10] This cannot be improved: for a d-regular graph with girth at least 2d, the high girth forces any planar subgraph to be sparse, causing its thickness to be exactly ⌈d/2⌉.[11]

Sulanke's nine-color Earth–Moon map, with adjacencies described by the join of a 6-vertex complete graph and 5-vertex cycle graph (center). Because the adjacencies in this graph are the union of those in two planar maps (left and right) it has thickness two.

Graphs of thickness t with n vertices have at most t(3n−6) edges. Because this gives them average degree less than 6t, their degeneracy is at most 6t−1 and their chromatic number is at most 6t. Here, the degeneracy can be defined as the maximum, over subgraphs of the given graph, of the minimum degree within the subgraph. In the other direction, if a graph has degeneracy D then its arboricity and thickness are at most D. One can find an ordering of the vertices of the graph in which each vertex has at most D neighbors that come later than it in the ordering, and assigning these edges to D distinct subgraphs produces a partition of the graph into D trees, which are planar graphs.

Even in the case t=2, the precise value of the chromatic number is unknown; this is Gerhard Ringel's Earth–Moon problem. An example of Thom Sulanke shows that, for t=2, at least 9 colors are needed.[12]

Thickness is closely related to the problem of simultaneous embedding.Cite error: Closing </ref> missing for <ref> tag

Computational complexity

It is NP-hard to compute the thickness of a given graph, and NP-complete to test whether the thickness is at most two.[13] However, the connection to arboricity allows the thickness to be approximated to within an approximation ratio of 3 in polynomial time.

References

  1. ↑ Tutte, W. T. (1963), "The thickness of a graph", Indag. Math. 66: 567–577, doi:10.1016/S1385-7258(63)50055-9 .
  2. ↑ Mäkinen, Erkki; Poranen, Timo (2012), "An annotated bibliography on the thickness, outerthickness, and arboricity of a graph", Missouri J. Math. Sci. 24 (1): 76–87, doi:10.35834/mjms/1337950501, http://projecteuclid.org/euclid.mjms/1337950501 
  3. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named duncan2009
  4. ↑ 4.0 4.1 Cite error: Invalid <ref> tag; no text was provided for refs named mos
  5. ↑ (Mutzel Odenthal), Theorem 3.2.
  6. ↑ Alekseev, V. B.; Gončakov, V. S. (1976), "The thickness of an arbitrary complete graph", Mat. Sb., New Series 101 (143): 212–230, doi:10.1070/SM1976v030n02ABEH002267, Bibcode: 1976SbMat..30..187A .
  7. ↑ (Mutzel Odenthal), Theorem 3.4.
  8. ↑ Beineke, Lowell W.; Harary, Frank; Moon, John W. (1964), "On the thickness of the complete bipartite graph", Proc. Cambridge Philos. Soc. 60 (1): 1–5, doi:10.1017/s0305004100037385, Bibcode: 1964PCPS...60....1B .
  9. ↑ Dean, Alice M.; Hutchinson, Joan P.; Scheinerman, Edward R. (1991), "On the thickness and arboricity of a graph", Journal of Combinatorial Theory, Series B 52 (1): 147–151, doi:10.1016/0095-8956(91)90100-X .
  10. ↑ Halton, John H. (1991), "On the thickness of graphs of given degree", Information Sciences 54 (3): 219–238, doi:10.1016/0020-0255(91)90052-V 
  11. ↑ Sýkora, Ondrej; Székely, László A.; Vrt'o, Imrich (2004), "A note on Halton's conjecture", Information Sciences 164 (1–4): 61–64, doi:10.1016/j.ins.2003.06.008 
  12. ↑ "To the Moon and beyond", Graph Theory: Favorite Conjectures and Open Problems, II, Problem Books in Mathematics, Springer International Publishing, 2018, pp. 115–133, doi:10.1007/978-3-319-97686-0_11 
  13. ↑ Mansfield, Anthony (1983), "Determining the thickness of graphs is NP-hard", Mathematical Proceedings of the Cambridge Philosophical Society 93 (1): 9–23, doi:10.1017/S030500410006028X, Bibcode: 1983MPCPS..93....9M .



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