Stockmeyer图上的Seurat对策
Seurat games on Stockmeyer graphs
摘要：
我们定义了一类顶点着色博弈，博弈者是$\forall$和$\exists$ .这些游戏源于代数逻辑中一个长期存在的开放问题。证明了存在一个自然数$n$，使得当$G\not\cong H$时，$\forall$在$n$颜色的博弈中总是有获胜策略.这与图的重构猜想和有向图的度相关重构猜想有关。我们证明了重构猜想蕴含着我们对图的与$n=3$的博弈猜想，并且对于度相关的重构猜想和我们对有向图的猜想也是如此。我们表明（对于任何$k<ω$），2色游戏可以区分某些非同构对的图，不能区分的$k$维Weisfeiler-Leman算法。我们还表明，2-色游戏可以区分的非同构对的图在家庭中定义的Stockmeyer作为反例，原来的有向图重建猜想。
英文摘要：
We define a family of vertex colouring games played over a pair of graphs or digraphs $(G,H)$ by players $\forall$ and $\exists$. These games arise from work on a longstanding open problem in algebraic logic. It is conjectured that there is a natural number $n$ such that $\forall$ always has a winning strategy in the game with $n$ colours whenever $G\not\cong H$. This is related to the reconstruction conjecture for graphs and the degree-associated reconstruction conjecture for digraphs. We show that the reconstruction conjecture implies our game conjecture with $n=3$ for graphs, and the same is true for the degree-associated reconstruction conjecture and our conjecture for digraphs. We show (for any $k<ω$) that the 2-colour game can distinguish certain non-isomorphic pairs of graphs that cannot be distinguished by the $k$-dimensional Weisfeiler-Leman algorithm. We also show that the 2-colour game can distinguish the non-isomorphic pairs of graphs in the families defined by Stockmeyer as counterexamples to the original digraph reconstruction conjecture.
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