Evasiveness猜想与2p点图
The Evasiveness Conjecture and Graphs on 2p Vertices
摘要：
对顶点数为素数幂的图和六个顶点的图的性质证明了Evavesiveness猜想。十个顶点的案子还没解决。本文通过研究图在$2p$顶点上的自同构群的大小来估计单调非回避图性质的欧拉示性数，并得到了单调非回避图性质必须满足的条件。我们也这样做，通过奥利弗群，并给出一些下界的单纯复形的维数与一个非平凡的单调非回避图形属性。我们应用我们的结果，在十个顶点的情况下，得到潜在的反例的回避猜想的条件。
英文摘要：
The Evasiveness conjecture have been proved for properties of graphs on a prime-power number of vertices and the six vertices case. The ten vertices case is still unsolved. In this paper we study the size of the automorphism group of a graph on $2p$ vertices to estimate the Euler characteristic of monotone non-evasive graph properties and get some conditions such graph properties must satisfy. We also do this by means of Oliver groups and give some lower bounds for the dimension of the simplicial complex associated to a nontrivial monotone non-evasive graph property. We apply our results to graphs on ten vertices to get conditions on potential counterexamples to the evasiveness conjecture in the ten vertices case.
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