Automorphism groups of graphs of bounded Hadwiger number

Martin Grohe, Pascal Schweitzer, Daniel Wiebking

We determine the structure of automorphism groups of finite graphs of bounded Hadwiger number. This in particular settles three of Babai's conjectures from the 1980s. The first one states that the order of non-alternating, non-abelian composition factors for automorphism groups of graphs of bounded Hadwiger number is bounded. The second one, the subcontraction conjecture, states that a non-trivial minor-closed graph class represents only finitely many non-abelian simple groups. And the third one states that if the order of such a group does not have small prime factors, then the group is obtained by iterated wreath and direct products from abelian groups. Our proof includes a structural analysis of finite edge-transitive graphs.

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