关于有向图的乘法性
On multiplicativity of directed graphs
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中文总结 AI 辅助
研究有向图范畴中乘法图，以可推同态为态射，给出有向二分图、有向循环和传递竞赛图的乘法性刻画，找到新的非乘法有向图类，还解决了关于指数有向图存在性的开放问题。
中文摘要 AI 辅助
图范畴是一个范畴，其中一组图或类似结构（如实图、有向图、符号图等）作为对象，适当的同态概念作为态射。无向图和有向图范畴中乘法对象的刻画是重要的开放问题。虽然Shitov最近对声称所有完全图都是乘法的Hedetniemi猜想的反驳在无向图乘法性研究上取得了突破，但无向图乘法性的刻画仅对循环、圆团块\(K_{n/k}\)（其中\({n/k} \in (2,4]\)）、完全图以及每条边最多是一个\(4\) - 循环一部分的图已知。类似地，对于给定有向图是否为乘法图仅对某些有向路径、有向循环和传递竞赛图已知。我们研究以可推同态作为态射的有向图范畴中的乘法图。我们给出了有向二分图、有向循环和传递竞赛图的完全乘法性刻画。结果，我们在通常的有向图范畴中找到了新的（无限）类非乘法有向图。我们还解决了Das等人（CALDAM 2026）提出的关于相对于可推同态的指数有向图存在性的一个开放问题，并将我们的解决方案用作证明工具。
英文摘要
A graph category is a category with a set of graphs or similar structures (such as, directed graphs, signed graphs, etc.) playing the role of objects, and an appropriate notion of homomorphism playing the role of morphisms. The characterization of multiplicative objects are important open problems in categories of undirected and directed graphs. While the recent disproving of the Hedetniemi's conjecture due to Shitov (Ann. Math. 2019), which claimed that all complete graphs are multiplicative, provided a breakthrough in the study of multiplicative undirected graphs, the characterization of multiplicative undirected graphs remains known only for cycles, circular cliques $K_{n/k}$ where ${n/k} \in (2,4]$, complete graphs, and graphs whose each edge is part of at most one $4$-cycle. Similarly, whether a given directed graph is multiplicative or not is known only for some oriented paths, oriented cycles, and transitive tournaments. We study multiplicative graphs in the category of directed graphs where pushable homomorphism plays the role of morphism. We provide full multiplicativity characterization for directed bipartite graphs, oriented cycles, and transitive tournaments. As a consequence we find new (infinite) classes of non-multiplicative directed graphs in the usual directed graphs category. We also resolve an open question posed by Das \textit{et al.} (CALDAM 2026) related to the existence of exponential directed graphs with respect to pushable homomorphisms, and use our solution as a tool for our proofs.
