The Evasiveness Conjecture and Graphs on 2p Vertices

TL;DR

This paper investigates the evasiveness conjecture for graphs on 2p vertices, providing bounds and conditions related to automorphism groups, Oliver groups, and simplicial complexes, especially focusing on the unresolved case of ten vertices.

Contribution

It introduces new bounds and conditions on automorphism groups and simplicial complexes for non-evasive graph properties, advancing understanding of the ten vertices case.

Findings

Lower bounds for automorphism group sizes

Conditions on simplicial complex dimensions

Constraints on potential counterexamples for ten vertices

Abstract

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.