AI 中文总结
本文构造连通图反例，反驳强罗伯森猜想及其浸入版本，证明排除特定子式或浸入的图类同态计数仍可确定目标同态数，并给出无限族显式反例。方法基于模等价与素数阶自同构。
AI 中文摘要
我们反驳了强罗伯森猜想，该猜想断言，在某个对 minors（子式）和不相交并封闭的图类之外添加任何图，都会严格增加来自该类的同态计数的区分能力。更精确地说，我们构造了连通图 $H$，使得排除 $H$ 作为 minor 的图的计数决定了从 $H$ 到任何目标图的同态数量。我们还反驳了以 immersions（浸入）替代 minors 的类似猜想。我们给出了显式的无限族排除图，包括立方二分图，它们为这两种关系提供了反例。证明引入了一种从模等价推导精确同态计数依赖性的方法。我们利用图的素数阶自同构（这些图排除其轨道商作为 minors 或 immersions）为无限多个素数获得了这些等价关系。
英文摘要
We refute the Strong Roberson Conjecture, which asserts that adding any graph outside a class closed under minors and disjoint unions strictly increases the distinguishing power of homomorphism counts from that class. More precisely, we construct connected graphs $H$ for which counts from graphs excluding $H$ as a minor determine the number of homomorphisms from $H$ to any target graph. We also refute the analogous conjecture with immersions in place of minors. We give explicit infinite families of excluded graphs, including cubic bipartite graphs that yield counterexamples for both relations. The proof introduces a method for deriving exact homomorphism count dependence from modular equivalences. We obtain these equivalences for infinitely many primes using prime-order automorphisms of graphs that exclude their orbit quotients as minors or immersions.
Comments22 pages
