A Min-Max theorem about the Road Coloring Conjecture

The Road Coloring Conjecture is an old and classical conjecture posed in Adler and Weiss (1970); Adler et al. (1977). Let G be a strongly connected digraph with uniform out-degree 2. The Road Coloring Conjecture states that, under a natural (necessary) condition that G is “aperiodic”, the edges of G can be colored red and blue such that “universal driving directions” can be given for each vertex. More precisely, each vertex has one red and one blue edge leaving it, and for any vertex v there exists a sequence sv of reds and blues such that following the sequence from any starting vertex in G ends precisely at the vertex v. We first generalize the conjecture to a min-max conjecture for all strongly connected digraphs. We then generalize the notion of coloring itself. Instead of assigning exactly one color to each edge we allow multiple colors to each edge. Under this relaxed notion of coloring we prove our generalized Min-Max theorem. Using the Prime Number Theorem (PNT) we further show that the number of colors needed for each edge is bounded above by O(log n/ log log n), where n is the number of vertices in the digraph.