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A connected graph whose automorphism group acts transitively on the edges andvertices, but not on the set of ordered pairs of adjacent vertices of the graphis called half-arc-transitive. It is well known that the valence of ahalf-arc-transitive graph is even and at least four. Several infinite familiesof half-arc-transitive graphs of valence four are known, however, in all exceptfour of the known specimens, the vertex-stabiliser in the automorphism group isabelian. The first example of a half-arc-transitive graph of valence four andwith a non-abelian vertex-stabiliser was described in [Conder andMaru\v{s}i\v{c}, A tetravalent half-arc-transitive graph with non-abelianvertex stabilizer, J. Combin. Theory Ser. B 88 (2003) 67--76]. This example has$10752$ vertices and vertex-stabiliser isomorphic to the dihedral group oforder $8$. In this paper, we show that no such graphs of smaller order exist,thus answering a frequently asked question.

Smallest tetravalent half-arc-transitive graphs with the vertex-stabiliser isomorphic to the dihedral group of order $8$