On Hamiltonian cycle systems with a nice automorphism group

Transcription

1 On Hamiltonian cycle systems with a nice automorphism group Francesca Merola Università Roma Tre April Hamiltonian cycle systems set B = {C,..., C s } of n-cycles of Γ such that the edges E(C ),... E(C s ) form a partition of E(Γ)

2 Hamiltonian cycle systems set B = {C,..., C s } of n-cycles of Γ such that the edges E(C ),... E(C s ) form a partition of E(Γ) or K n I, I a -factor, n even Hamiltonian cycle systems set B = {C,..., C s } of n-cycles of Γ such that the edges E(C ),... E(C s ) form a partition of E(Γ) or K n I, I a -factor, n even

3 Hamiltonian cycle systems set B = {C,..., C s } of n-cycles of Γ such that the edges E(C ),... E(C s ) form a partition of E(Γ) or K n I, I a -factor, n even Hamiltonian cycle systems set B = {C,..., C s } of n-cycles of Γ such that the edges E(C ),... E(C s ) form a partition of E(Γ) or K n I, I a -factor, n even

4 more generally: graph decomposition Let G be a graph with vertex set V (G) and edge set E(G) a decomposition of G is a set of subgraphs of G whose edge sets partition the edge set of G a (G, Γ)-decomposition is a decomposition in which the subgraphs are all isomorphic to Γ A (K 7, K )-decomposition [or a (K 7, C )-decomposition] the vertices of the K s are {,, }, {,, }, {,, }, {,, }, {,, }, {,, }, {,, }

5 existence - Walecki s construction n odd n even removed -factor: [, ], [, ], [, ], [, ] regular cycle systems a HCS is regular if there is an automorphism group G of Γ acting sharply transitively on the vertices of Γ (so we identify V (Γ) with G) permuting the cycles of B it is called cyclic if G is a cyclic group we shall call it dihedral if G is a dihedral group to construct regular cs it is enough to give its base cycles i.e. a set F of representatives for the G-orbits of the cycles

6 cyclic HCS the existence of hamiltonian cyclic CS is completely settled n odd: a cyclic cs for K n iff n and n p α (p an odd prime and α > ) [Buratti, Del Fra ()] n even, a cyclic cs for K n I, iff n, (mod ) and n p α (p an odd prime and α ) [Gavlas-Jordon, Morris ()] a cyclic HCS for K I removed -factor: [, ] [, 7] [, ] [, 9] [, ] [, ]

7 non-hamiltonian CS we may consider a decomposition of Γ into k-cycles (k < n) obvious necessary conditions for existence of a k-cs of Γ k V (Γ); the vertices of Γ have even degree; k E(Γ) when Γ = K n, n odd or K n I, I a -factor, n even these conditions are also sufficient Alspach and Gavlas-Jordon () for n and k both odd or both even, Šajna () in the remaining cases but for cyclic k-cs known only for n, k (mod k) dihedral HCS consider a hamiltonian cycle system regular under the dihedral group D n n even, n = m the graph is K m I, I a -factor Theorem (M. Buratti, FM, ) There is a dihedral (K m I, C m )-design for all even m. There is a dihedral hamiltonian cycle system for K m I, m an odd integer, iff m has at least two distinct prime factors there is a suitable e such that p (mod e ) for all prime factors p of m and the number of those (counted with their respective multiplicities) such that p (mod e+ ) is even. note that, in the cyclic case, there is no hamiltonian cycle system when n (mod ) the odd integers < for which there is a dihedral HCS are,,, 7,, 9, 77, 9

8 a dihedral HCS for K I D = {, x, x, x, y, xy, x y, x y}, (x = y =, yxy = x ) x x y x x x y x x x x y x y xy x x y xy y x y x y xy x x y xy y x x the removed -factor is [, y] [x, x y] [x, xy] [x, x y] x y x y sharply vertex-transitive HCS a sharply vertex-transitive HCS(n) exists for n odd, iff n p α p prime, α > for n even, iff n pα p prime, α this comes from the cyclic and dihedral results plus some extra cases when n (mod ) (Buratti, unpublished) but - for which groups G is there a HCS which is sharply vertex transitive under G? not known and hard

9 -rotational HCS(n=k+) symmetric terrace of a binary ( involution, λ) group an ordering of the elts of G of the form (g, g,..., g k, g k + λ,..., g + λ, g + λ) such that {±(g i+ g i, i k )} = G \ {, λ} e.g. for Z (,, 7,,,,, ) Bailey-Ollis-Preece () ST -rotational HCS under G Anderson-Ihrig (99) G Q and soluble G has ST G non soluble - open Buratti-Rinaldi-Traetta () -rotational HCS under G ST BOP+BRT for k up to isomorphism at least k -rotational HCS(k + ) full automorphism group want to determine the groups G for which there is an HCS whose full automorphism group is G there is a very recent result by Buratti, Lovegrove, Traetta necessary condition: G AGL(, p) p prime or G is binary or G is odd also sufficient with the possible exception of G binary and non soluble.