Road coloring theorem
In graph theory the road coloring theorem, known until recently as the road coloring conjecture, deals with synchronized instructions. The issue involves whether by using such instructions, one can reach or locate an object or destination from any other point within a network (which might be a representation of city streets or a maze). In the real world, this phenomenon would be as if you called a friend to ask for directions to his house, and he gave you a set of directions that worked no matter where you started from. This theorem also has implications in symbolic dynamics.
Example and intuition
The image to the right shows a directed graph on eight vertices in which each vertex has out-degree 2. (Each vertex in this case also has in-degree 2, but that is not necessary for a synchronizing coloring to exist.) The edges of this graph have been colored red and blue to create a synchronizing coloring.
For example, consider the vertex marked in yellow. No matter where in the graph you start, if you traverse all nine edges in the walk "blue-red-red—blue-red-red—blue-red-red", you will end up at the yellow vertex. Similarly, if you traverse all nine edges in the walk "blue-blue-red—blue-blue-red—blue-blue-red", you will always end up at the vertex marked in green, no matter where you started.
The road coloring theorem states that for a certain category of directed graphs, it is always possible to create such a coloring.
Let G be a finite, strongly connected, directed graph where all the vertices have the same out-degree k. Let A be the alphabet containing the letters 1, ..., k. A synchronizing coloring (also known as a collapsible coloring) in G is a labeling of the edges in G with letters from A such that (1) each vertex has exactly one outgoing edge with a given label and (2) for every vertex v in the graph, there exists a word w over A such that all paths in G corresponding to w terminate at v.
For such a coloring to exist at all, it is necessary that G be aperiodic. The road coloring theorem states that aperiodicity is also sufficient for such a coloring to exist. Therefore, the road coloring problem can be stated briefly as:
- Every finite strongly connected aperiodic directed graph of uniform out-degree has a synchronizing coloring.
Previous partial results
Previous partial or special-case results include the following:
- If G is a finite strongly connected aperiodic directed graph with no multiple edges, and G contains a simple cycle of prime length which is a proper subset of G, then G has a synchronizing coloring. (O'Brien 1981)
- If G is a finite strongly connected aperiodic directed graph (multiple edges allowed) and every vertex has the same in-degree and out-degree k, then G has a synchronizing coloring. (Kari 2003)
- Adler, R.L.; Weiss, B. (1970), Similarity of automorphisms of the torus, Memoires of the American Mathematical Society 98.
- Hegde, Rajneesh; Jain, Kamal (2005), "A min-max theorem about the road coloring conjecture", Proc. EuroComb 2005 (PDF), Discrete Mathematics & Theoretical Computer Science, pp. 279–284.
- Kari, Jarkko (2003), "Synchronizing finite automata on Eulerian digraphs", Theoretical Computer Science 295 (1-3): 223–232, doi:10.1016/S0304-3975(02)00405-X.
- O'Brien, G. L. (1981), "The road-colouring problem", Israel Journal of Mathematics 39 (1–2): 145–154, doi:10.1007/BF02762860.
- Trahtman, Avraham N. (2009), "The road coloring problem", Israel Journal of Mathematics 172 (1): 51–60, arXiv:0709.0099, doi:10.1007/s11856-009-0062-5.