The Road Coloring for Mapping on k States(withdrawn)

TL;DR

This paper discusses the problem of road coloring in directed strongly connected graphs with uniform outdegree, focusing on automaton mappings induced by edge colorings, aiming to understand the conditions for synchronizing colorings.

Contribution

It introduces new insights into the road coloring problem for graphs with uniform outdegree and automaton mappings, extending previous theoretical frameworks.

Findings

Conditions for the existence of synchronizing colorings

Characterization of automaton mappings induced by edge colorings

Extensions to the theory of deterministic automata

Abstract

Let $\Gamma$ be directed strongly connected finite graph of uniform outdegree (constant outdegree of any vertex) and let some coloring of edges of $\Gamma$ turn the graph into deterministic complete automaton. Let the word $s$ be a word in the alphabet of colors (considered also as letters) on the edges of $\Gamma$ and let $\Gamma s$ be a mapping of vertices $\Gamma$.