On totally synchronizing graphs

2026-07-19

AI Summary

This study investigates whether every strongly connected directed graph with constant out-degree \(k\) is synchronizing under any edge coloring—that is, whether it is a “completely synchronizing” digraph. Integrating tools from graph theory, automata theory, group actions, strong aggregability of Markov chains, and algebraic graph theory, the work reveals structural and symmetry constraints on such graphs: it proves that their automorphism groups contain no semiregular elements, introduces the novel notion of “totally simple” digraphs, and establishes a sufficient condition based on the Perron–Frobenius eigenvector. Key contributions include explicit constructions of counterexamples, symmetry-based obstructions to complete synchronizability, and proofs that both the complete synchronizability problem and the existence of a non-synchronizing coloring are NP-complete.

Abstract

A coloring of a finite $k$-out directed graph $G$ is viewed as a deterministic complete automaton with state set $V(G)$. The graph $G$ is called \emph{totally synchronizing} if every coloring is synchronizing. We prove that total synchronization imposes strong restrictions on symmetry: if $G$ is strongly connected and totally synchronizing, then $Aut(G)$ contains no semiregular element; in particular, if $|Aut(G)|$ is divisible by a prime $p>k$, then $G$ is not totally synchronizing. We then give general constructions of strongly connected $k$-out graphs with prescribed quotients and prescribed automorphism group that are \emph{not} totally synchronizing. On the quotient side, we relate graph congruences to strong lumpability of the uniform random walk on $G$ and introduce \emph{totally simple} graphs, characterized by the absence of nontrivial congruences. In this setting we obtain a Perron--Frobenius sufficient condition for total synchronization: a strongly connected non-lumpable graph whose integer Perron--Frobenius eigenvector admits at most one nontrivial equipartition is totally synchronizing. Finally, we show that deciding whether a primitive $k$-out graph admits a non-synchronizing coloring is NP-complete, resolving an open problem of Gusev--Szykuła, and prove NP-completeness of deciding whether a graph admits a nontrivial Eulerian lumping.

Problem

• totally synchronizing
• synchronizing coloring
• automorphism group
• strong lumpability
• NP-completeness

Innovation

• totally synchronizing graphs
• automorphism groups
• strong lumpability
• Perron–Frobenius eigenvector
• NP-completeness