Rajneesh Hegde ; Kamal Jain - A Min-Max theorem about the Road Coloring Conjecture

dmtcs:3455 - Discrete Mathematics & Theoretical Computer Science, January 1, 2005, DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05) - https://doi.org/10.46298/dmtcs.3455
A Min-Max theorem about the Road Coloring ConjectureArticle

Authors: Rajneesh Hegde 1; Kamal Jain 2

The Road Coloring Conjecture is an old and classical conjecture e 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 $s_v$ of reds and blues such that following the sequence from $\textit{any}$ 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.


Volume: DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Section: Proceedings
Published on: January 1, 2005
Imported on: May 10, 2017
Keywords: road coloring,synchronization of automata,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]

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