Strong coloring


In graph theory, a strong coloring, with respect to a partition of the vertices into subsets of equal sizes, is a vertex coloring in which every color appears exactly once in every part. A graph is strongly k-colorable if, for each partition of the vertices into sets of size k, it admits a strong coloring. When the order of the graph G is not divisible by k, we add isolated vertices to G just enough to make the order of the new graph ' divisible by k. In that case, a strong coloring of ' minus the previously added isolated vertices is considered a strong coloring of G.
The strong chromatic number sχ of a graph G is the least k such that G is strongly k-colorable.
A graph is strongly k-chromatic if it has strong chromatic number k.
Some properties of sχ:
  1. sχ > Δ.
  2. sχ ≤ 3 Δ − 1.
  3. Asymptotically, sχ ≤ 11 Δ / 4 + o.
Here, Δ is the maximum degree.
Strong chromatic number was independently introduced by Alon and Fellows.

Related problems

Given a graph and a partition of the vertices, an independent transversal is a set U of non-adjacent vertices such that each part contains exactly one vertex of U. A strong coloring is equivalent to a partition of the vertices into disjoint independent-transversals. This is in contrast to graph coloring, which is a partition of the vertices of a graph into a given number of independent sets, without the requirement that these independent sets be transversals.
To illustrate the difference between these concepts, consider a faculty with several departments, where the dean wants to construct a committee of faculty members. But some faculty members are in conflict and will not sit in the same committee. If the "conflict" relations are represented by the edges of a graph, then:
  • An independent set is a committee with no conflict.
  • An independent transversal is a committee with no conflict, with exactly one member from each department.
  • A graph coloring is a partitioning of the faculty members into committees with no conflict.
  • A strong coloring is a partitioning of the faculty members into committees with no conflict and with exactly one member from each department. Thus this problem is sometimes called the happy dean problem.