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.
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.
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.
Gimbel, John; Straight, H. Joseph (1987), "Some topics in cochromatic theory", Graphs and Combinatorics, 3 (1): 255–265, doi:10.1007/BF01788548.
Jørgensen, Leif K. (1995), "Critical 3-cochromatic graphs", Graphs and Combinatorics, 11 (3): 263–266, doi:10.1007/BF01793013.
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), , Research report RRR 16-2000, Rutgers University Center for Operations Research, archived from the original on 2016-03-03, retrieved 2006-10-16.
March 13, 2023
cocoloring, graph, theory, cocoloring, graph, assignment, colors, vertices, such, that, each, color, class, forms, independent, complement, cochromatic, number, fewest, colors, needed, cocolorings, graphs, with, cochromatic, number, exactly, bipartite, graphs,. 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 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 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 amp Straight 1977 Jorgensen 1995 characterizes critical 3 cochromatic graphs while Fomin Kratsch amp 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 EditFomin 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 Gimbel John Straight H Joseph 1987 Some topics in cochromatic theory Graphs and Combinatorics 3 1 255 265 doi 10 1007 BF01788548 Jorgensen Leif K 1995 Critical 3 cochromatic graphs Graphs and Combinatorics 11 3 263 266 doi 10 1007 BF01793013 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 Retrieved from https en wikipedia org w index php title Cocoloring amp oldid 969376637, wikipedia, wiki, book, books, library,
