Abstract

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.

Highlights

  • Imagine a network of one-way roads between a set of cities, such that there are exactly two roads leaving each city

  • For every destination city, there is a sequence of reds and blues such that following the entire sequence from any starting city ends precisely in the destination city. (Note that the sequence must end on the city, not just pass through it.) For such a coloring to exist, two necessary conditions are (i) the network is strongly connected, and (ii) it is “aperiodic”

  • Multicolor Road Coloring Theorem: In this paper we prove a road coloring theorem which extends the Strong Road Coloring Conjecture (S-RCC) to all strongly connected digraphs under a suitably extended notion of coloring

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Summary

A Min-Max theorem about the Road Coloring Conjecture

To cite this version: Rajneesh Hegde, Kamal Jain. A Min-Max theorem about the Road Coloring Conjecture. 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb ’05), 2005, Berlin, Germany. pp.279-284. ￿hal-01184444￿. A Min-Max theorem about the Road Coloring Conjecture. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. We first generalize the conjecture to a min-max conjecture for all strongly connected digraphs. 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

Introduction
Problem Definitions
Theorem Statements and Proof Ideas
Discussion
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