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.
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.
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
triangle-free graph with maximum degree 4 (Gimbel & Hartman 2003) (Fiala et al. 2003),
comparability graph with maximum degree 4 (Ochem 2017),
line graph of a bipartite graph with maximum degree 4 (Gonçalves & Ochem 2009),
graph with girth 5 (Montassier & Ochem 2015).
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
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", Computing, 69 (3): 187–203, doi:10.1007/s00607-002-1461-1.
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.
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.
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