Lesson plan
Subcoloring
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 \chi_S(G) of a graph G is the fewest colors needed in any subcoloring of G . Subcoloring and subchromatic number were introduced by Michael O. Albertson, Robert E. Jamison, Stephen T. Hedetniemi, and Stephen C. Locke in 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, * comparability graph with maximum degree 4, * line graph of a bipartite graph with maximum degree 4, * or a graph with girth 5. The subchromatic number of a cograph can be computed in polynomial time. 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 .
Article summary
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 \chi_S(G) of a graph G is the fewest colors needed in any subcoloring of G . Subcoloring and subchromatic number were introduced by Michael O. Albertson, Robert E. Jamison, Stephen T. Hedetniemi, and Stephen C. Locke in 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, * comparability graph with maximum degree 4, * line graph of a bipartite graph with maximum degree 4, * or a graph with girth 5. The subchromatic number of a cograph can be computed in polynomial time. 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 .
Key claims
No claims recorded.
Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
No sources recorded.
Key claims and bibliography are generated from the same tracked claims and sources as the article page.