Most actions on regular trees are almost free

9 O ct 2 00 8 MOST ACTIONS ON REGULAR TREES ARE ALMOST FREE

Let T be a d-regular tree (d ≥ 3) and A = Aut(T ) its automorphism group. Let Γ be the group generated by n independent Haar-random elements of A. We show that almost surely, every nontrivial element of Γ has finitely many fixed points on T .

Mixing Actions of Countable Groups Are Almost Free

A measure preserving action of a countably in nite group Γ is called totally ergodic if every in nite subgroup of Γ acts ergodically. For example, all mixing and mildly mixing actions are totally ergodic. This note shows that if an action of Γ is totally ergodic then there exists a nite normal subgroup N of Γ such that the stabilizer of almost every point is equal to N. Surprisingly the proof r...

Regular polygons are most tolerant

Tree-minimal Graphs Are Almost Regular

For all fixed trees T and any graph G we derive a counting formula for the number N̂T (G) of homomorphisms from T to G in terms of the degree sequence of G. As a consequence we obtain that any n-vertex graph G with edge density p = p(n) n−1/(t−2), which contains at most (1 + o(1))pt−1nt copies of some fixed tree T with t ≥ 3 vertices must be almost regular, i.e., ∑ v∈V (G) |deg(v)− pn| = o(pn ).

Almost all Regular Graphs are Hamiltonian

In a previous paper the authors showed that almost all labelled cubic graphs are hamiltonian. In the present paper, this result is used to show that almost all r-regular graphs are hamiltonian for any fixed r ≥ 3, by an analysis of the distribution of 1-factors in random regular graphs. Moreover, almost all such graphs are r-edge-colourable if they have an even number of vertices. Similarly, al...