A non-optimal subcoloring with four colors. Merging the red and blue colors, and the green and yellow colors, produces a subcoloring with only two colors.

In graph theory, a subcoloring is an assignment of colors to a graph's vertices such that each color class induces a vertex disjoint union of cliques. That is, each color class should form a cluster graph.

The subchromatic number χS(G) of a graph G is the fewest colors needed in any subcoloring of G.

Subcoloring and subchromatic number were introduced by Albertson et al. (1989).

Every proper coloring and cocoloring of a graph are also subcolorings, so the subchromatic number of any graph is at most equal to the cochromatic number, which is at most equal to the chromatic number.

Subcoloring is as difficult to solve exactly as coloring, in the sense that (like coloring) it is NP-complete. More specifically, the problem of determining whether a planar graph has subchromatic number at most 2 is NP-complete, even if it is a

The subchromatic number of a cograph can be computed in polynomial time (Fiala et al. 2003). For every fixed integer r, it is possible to decide in polynomial time whether the subchromatic number of interval and permutation graphs is at most r (Broersma et al. 2002).

References

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  • Albertson, M. O.; Jamison, R. E.; Hedetniemi, S. T.; Locke, S. C. (1989), "The subchromatic number of a graph", Discrete Mathematics, 74 (1–2): 33–49, doi:10.1016/0012-365X(89)90196-9.
  • Broersma, Hajo; Fomin, Fedor V.; Nesetril, Jaroslav; Woeginger, Gerhard (2002), "More About Subcolorings" (PDF), Computing, 69 (3): 187–203, doi:10.1007/s00607-002-1461-1, S2CID 24777938.
  • Fiala, J.; Klaus, J.; Le, V. B.; Seidel, E. (2003), "Graph Subcolorings: Complexity and Algorithms", SIAM Journal on Discrete Mathematics, 16 (4): 635–650, CiteSeerX 10.1.1.3.183, doi:10.1137/S0895480101395245.
  • Gimbel, John; Hartman, Chris (2003), "Subcolorings and the subchromatic number of a graph", Discrete Mathematics, 272 (2–3): 139–154, doi:10.1016/S0012-365X(03)00177-8.
  • Gonçalves, Daniel; Ochem, Pascal (2009), "On star and caterpillar arboricity", Discrete Mathematics, 309 (11): 3694–3702, doi:10.1016/j.disc.2008.01.041.
  • Montassier, Mickael; Ochem, Pascal (2015), "Near-Colorings: Non-Colorable Graphs and NP-Completeness", Electronic Journal of Combinatorics, 22 (1): #P1.57, arXiv:1306.0752, doi:10.37236/3509, S2CID 59507.
  • Ochem, Pascal (2017), "2-subcoloring is NP-complete for planar comparability graphs", Information Processing Letters, 128: 46–48, arXiv:1702.01283, doi:10.1016/j.ipl.2017.08.004, S2CID 22108461.

📚 Artikel Terkait di Wikipedia

Graph coloring

then every path between them contain a vertex with color greater than i Subcoloring An improper vertex coloring where every color class induces a union of

List of graph theory topics

edge-coloring Perfect graph Ramsey's theorem Sperner's lemma Strong coloring Subcoloring Tait's conjecture Total coloring Uniquely colorable graph Path (graph

Cluster graph

number of different types of countably infinite homogeneous graphs. A subcoloring of a graph is a partition of its vertices into induced cluster graphs

Cocoloring

which each color class must be an independent set) and stronger than for subcoloring (in which each color class must be a disjoint union of cliques), it follows