Regular groups of automorphisms of cubic graphs

Abstract

W.T. Tutte showed that if G is an arc transitive connected cubic graph then the automorphism group of G is in fact regular on s-arcs for some s < 5. We analyze these arc transitive cubic graphs using the unifying concepts of the infinite cubic tree, f3' and coverings. We are able to answer a large number of questions, open and otherwise. As an example, suppose G is a 4-arc transi tive cubic graph and the automorphism group of G contains a 1-regular subgroup, then G is a covering of Heawood's graph. Abstract W.T. Tutte showed that if G is an arc transitive connected cubic graph then the automorphism group of G is in fact regular on s-arcs for some s < 5. We analyze these arc transitive cubic graphs -> using the unifying concepts of the infinite cubic tree, r 3 , and coverings. We are able to answer a large number of questions, open and otherwise. As an example, suppose G is a 4-arc transitive cubic graph and the automorphism group of G contains a l-regular subgroup, then G is a covering of Heawood1s graph. Regular Groups of Automorphisms of Cubic Graphs v Dragomir Z. Djokovit and Gary L. Miller O. Introduction. This paper is concerned with s-regular groups of automorphisms of connected cubic graphs. (See next section for definitions.) The subject was started by W.T. Tutte [13] by proving the basic fact that s s 5 in the case of finite cubic graphs. (See also [15]). He also proved in that case that if the graph is l-transitive then it must be s-regular for some s, 1 s s s 5. With slight modifications these results remain valid for infinite cubic graphs. Let G be a connected cubic graph and let {u.y} be an edge of G. Let A be an s-regular subgroup of Aut{G), Then we associate to A the amalgam (A(u),A[u,v]) where A{u) is the fixer of u in A and A[u,v] is the stabilizer of . {u,v} in A. This amalgam is independent (up to isomorphism) of the chosen edge {u,v}. Moreover, if s = 1,3, or 5 this amalgam is even independent of the choice of G and A and if s = 2 or 4 there are two possible amalgams in each case. Hence we obtain seven possible types of. s-regular groups (ls-ss5): 1',2 1 ,2",3',4',4", and 51. The group of type Sl is s-regular and has an involution which flips an edge. The groups of type s" are s-regular and they do not possess any involution flipping an edge. Let r3 be the cubic tree. We show that Aut(r 3 ) contains regular subgroups of all seven types, moreover two regular subgroups of the same type are conjugate. -1 ... 3 Moreover, in this way we obtain all such pairs (G,B) (up to isomorphism).This correspondence N~ (GN,A N ) is either one -one or two -one, i.e., the same pair can be obtained from at most two normal subgroups •. Hence the problem of classifying pairs (G,B) of the given type reduces to the problem of classifying normal subgroups of A.This latter problem is still far from its solution. For instance, if we consider the type l' then A is the modular group and in spite of an enormous number of papers on normal subgroups of it (see the references, in [9]) the final answer is still lacking. It is e...

Existence, covolumes and infinite generation of lattices for Davis complexes

“…Choose vertices s 1 and s 2 of L at distance two in L if n = 3, 4, and at distance three in L if n = 5. Then by Propositions 3-5 of Djoković-Miller [12], for i = 1, 2 there are involutions α i ∈ Aut(L) such that α i fixes the star of s 3−i in L, and α i (s i ) = s i is not adjacent to s i . Thus if m is even, the Main Theorem applies to G = Aut(Σ).…”

“…We describe several infinite families of examples of Davis complexes Σ to which our results apply in Section 5 below. To establish these applications, we use properties of spherical buildings in [25], and some results of graph theory from [12]. In two dimensions, examples include the Fuchsian buildings considered in [29], and some of the highly symmetric Platonic polygonal complexes investigated bý Swi ֒ atkowski [27].…”

Tetravalent one-regular graphs of order 2pq

“…By elementary group theory, up to isomorphism, there are three nonsolvable groups of order 120 which are SL(2, 5), A 5 × Z 2 and S 5 . Suppose G = SL (2,5 …”

“…The first example of cubic one-regular graph was constructed by Frucht [10] with 432 vertices, and much subsequent work was done in this line as part of a more general problem dealing with the investigation of cubic arc-transitive graphs (see [4][5][6][7][8][9]26]). Tetravalent one-regular graphs have also received considerable attention.…”

Tetravalent one-regular graphs of order 4p2

“…The first example of a cubic one-regular graph was constructed by Frucht [21]. Further research in cubic one-regular graphs has been part of a more general project dealing with the investigation of cubic arc-transitive graphs (see [9,15,[17][18][19][20]31]). Tetravalent one-regular graphs have also received considerable attention.…”