关于完全同步图
On totally synchronizing graphs
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中文总结 AI 辅助
研究有限\(k\)出有向图的完全同步性，证明其对对称性的限制，给出非完全同步图的构造，关联图同余与随机游走合并性，得到完全同步条件，证明判定本原\(k\)出图非同步着色及图的非平凡欧拉合并性问题是NP完全的。
中文摘要 AI 辅助
有限\(k\)出有向图\(G\)的一种着色被视为状态集为\(V(G)\)的确定性完全自动机。若每种着色都是同步的，则图\(G\)称为完全同步的。我们证明完全同步对对称性有严格限制，若\(G\)是强连通且完全同步的，则\(Aut(G)\)不包含半正则元素。接着给出具有规定商和规定自同构群且非完全同步的强连通\(k\)出图的一般构造。在商方面，将图同余与\(G\)上均匀随机游走的强合并性相关联并引入完全简单图。还得到完全同步的一个Perron - Frobenius充分条件。最后表明判定一个本原\(k\)出图是否有非同步着色是NP完全的，解决了Gusev - Szykuła的一个开放问题，并证明判定一个图是否有非平凡欧拉合并性是NP完全的。
英文摘要
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.
