Vertex-Transitive Graphs: From Semiregular To Simplicial Automorphisms

When dealing with symmetry of combinatorial objects { or in any other setting for that matter { onewill inexorably come across two different kinds of nonidentity automorphisms of these objects: those fixingas large as possible subset of points, on the one hand, and those fixing no points at all on the other. it isthe second of these two possibilities that is of interest here. such automorphisms are called derangements.by a theorem of fein, kantor and schacher [3, theorem 1], every transitive permutation group containsa derangement of prime power order. if we were to replace the requirement that a derangement is of‘prime power order’ with that of being of ‘prime order’ the result would not longer be true. such groups,although rare, do exist and are called elusive, see [2].existence of a derangement of prime order in a transitive permutation group is equivalent to theexistence of an element with all of its cycles in its cycle decomposition of the same length. such anelement is called semiregular and the conjecture regarding existence of such elements in automorphismgroups of vertex-transitive (di)graphs (and more general in 2-closed groups) is usually referred to as thesemiregularity problem (alternatively, the polycirculant conjecture), see [1, 5].in this lecture, i will discuss some recent developments in regards to a special case of the semiregularity problem regarding existence of simplicial automorphisms, that is, semiregular automorphisms whosequotient multigraphs are simple graphs. when dealing with structural properties of vertex-transitivegraphs, reductions via such automorphisms are a viable alternative in absence of normal subgroups. iwill give a special emphasis to existence of simplical automorphisms in cubic arc-transitive graphs with aprimitive automorphism group [4]