Cocoloring with 3 colors (upper left figure): a proper 3-coloring of this graph is impossible. The blue subgraph forms a clique (bottom right figure), while the red and green subgraphs form cliques on the graph's complement.

In graph theory, a cocoloring of a graph G is an assignment of colors to the vertices such that each color class forms an independent set in G or in the complement of G. The cochromatic number z(G) of G is the fewest colors needed in any cocolorings of G. The graphs with cochromatic number 2 are exactly the bipartite graphs, complements of bipartite graphs, and split graphs.

As the requirement that each color class be a clique or independent is weaker than the requirement for coloring (in 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 that the cochromatic number of G is less than or equal to the chromatic number of G, and that it is greater than or equal to the subchromatic number of G.

Cocoloring was named and first studied by Lesniak & Straight (1977). Jørgensen (1995) characterizes critical 3-cochromatic graphs, while Fomin, Kratsch & Novelli (2002) describe algorithms for approximating the cochromatic number of a graph. Zverovich (2000) defines a class of perfect cochromatic graphs, analogous to the definition of perfect graphs via graph coloring, and provides a forbidden subgraph characterization of these graphs.

References

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  • Fomin, Fedor V.; Kratsch, Dieter; Novelli, Jean-Christophe (2002), "Approximating minimum cocolourings", Inf. Process. Lett., 84 (5): 285–290, doi:10.1016/S0020-0190(02)00288-0, S2CID 17733740.
  • Gimbel, John; Straight, H. Joseph (1987), "Some topics in cochromatic theory", Graphs and Combinatorics, 3 (1): 255–265, doi:10.1007/BF01788548, S2CID 8218390.
  • Jørgensen, Leif K. (1995), "Critical 3-cochromatic graphs", Graphs and Combinatorics, 11 (3): 263–266, doi:10.1007/BF01793013, S2CID 38851896.
  • Lesniak, L.; Straight, H. J. (1977), "The cochromatic number of a graph", Ars Combinatoria, 3: 39–46.
  • Straight, H. J. (1979), "Cochromatic number and the genus of a graph", Journal of Graph Theory, 3 (1): 43–51, doi:10.1002/jgt.3190030106.
  • Zverovich, Igor V. (2000), Perfect cochromatic graphs, Research report RRR 16-2000, Rutgers University Center for Operations Research, archived from the original on 2016-03-03, retrieved 2006-10-16.

📚 Artikel Terkait di Wikipedia

Graph coloring

Motivated by task systems in which production proceeds in a cyclic way Cocoloring An improper vertex coloring where every color class induces an independent

Subcoloring

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

Glossary of graph theory

cograph is a graph produced by operations that include complementation; a cocoloring is a coloring in which each vertex induces either an independent set (as

List of graph theory topics

Museum guard problem Wheel graph Acyclic coloring Chromatic polynomial Cocoloring Complete coloring Edge coloring Exact coloring Four color theorem Fractional

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match as won by Starman and also included Bobby Zavala, Canelo Casas, Cocolores, Flamita, Fresbee, Halcon Magico, Kato, Principe Azteca, Robin, Trox,

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