University of Sydney Algebra Seminar

Reymond Akpanya

Friday 9 October, 12-1pm, in Carslaw 275

Regular Graphs with given Automorphism Groups

A classical theorem of Frucht states that every finite group occurs as the automorphism group of a finite graph. In this talk, we prove an embedded analogue for regular graphs of every degree at least three. In particular, for every integer \(d\geq3\) and every finite group \(G\), there exists a \(d\)-regular graph \(\Gamma\) together with a strong embedding \(\beta\) into a closed surface such that \(Aut(\Gamma)\cong Aut(\beta,\Gamma)\cong G\). Along the way, we identify an oversight in Sabidussi’s classical construction of regular graphs with prescribed automorphism groups. We give an alternative construction that resolves this issue and strengthens Sabidussi’s result by producing graphs that admit proper \(d\)-edge-colourings preserved by every graph automorphism. This is joint work with Meike Weiß (RWTH Aachen) and Tom Goertzen (University of Sydney).