Cocoloring

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

  • 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.

Content Disclaimer

Informasi ini disarikan dari Wikipedia dan disajikan kembali untuk tujuan edukasi. Konten tersedia di bawah lisensi CC BY-SA 3.0. Kami tidak bertanggung jawab atas ketidakakuratan data yang bersumber dari kontribusi publik tersebut.

  1. The information displayed on this website is sourced in part or in whole from Wikipedia and has been adapted for the purpose of restating it. We strive to provide accurate and relevant information, however:
  2. There is no guarantee of absolute accuracy. Wikipedia is an open, collaborative project that can be edited by anyone, so information is subject to change.
  3. It is not intended to constitute professional advice. The content displayed is for informational and educational purposes only. For important decisions (e.g., medical, legal, or financial), please consult a professional.
  4. Content copyright. Wikipedia is licensed under the Creative Commons Attribution-ShareAlike License (CC BY-SA). This means that content may be reused with appropriate attribution and shared under a similar license.
  5. Responsible use. Any risk arising from the use of information from this website is entirely the responsibility of the user.